3008:
8354:
7102:
3564:
2639:
2044:
1606:
7080:
47:
1647:
156:
3252:
6538:
6510:
6480:
6450:
5888:
5842:
5812:
5782:
5736:
3003:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\&=&((p\land \lnot q)\lor \lnot p)&\land &((p\land \lnot q)\lor q)\\&=&((p\lor \lnot p)\land (\lnot q\lor \lnot p))&\land &((p\lor q)\land (\lnot q\lor q))\\&=&(\lnot p\lor \lnot q)&\land &(p\lor q)\\&=&\lnot (p\land q)&\land &(p\lor q)\end{matrix}}}
7046:
7016:
6982:
6798:
6768:
6668:
6638:
5492:
5462:
204:
8748:
6284:
6254:
6224:
6190:
3559:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)&=&p{\overline {q}}+{\overline {p}}q\\&=&(p\lor q)&\land &(\lnot p\lor \lnot q)&=&(p+q)({\overline {p}}+{\overline {q}})\\&=&(p\lor q)&\land &\lnot (p\land q)&=&(p+q)({\overline {pq}})\end{matrix}}}
7397:. The XOR operation preserves randomness, meaning that a random bit XORed with a non-random bit will result in a random bit. Multiple sources of potentially random data can be combined using XOR, and the unpredictability of the output is guaranteed to be at least as good as the best individual source.
4234:
4991:
in 1890 (the original date is not definitely known, but almost certainly it is written after 1685; and 1890 is the publishing time). While both
Huntington in 1904 and Leibniz in 1890 used the symbol as an algebraic operation. Furthermore, Huntington in 1904 used the symbol as inclusive disjunction
7449:
XOR is also used to detect an overflow in the result of a signed binary arithmetic operation. If the leftmost retained bit of the result is not the same as the infinite number of digits to the left, then that means overflow occurred. XORing those two bits will give a "1" if there is an overflow.
3644:
4357:
The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphasized in a given context of discussion. In addition to the abbreviation "XOR", any of the following symbols may also be seen:
3129:
2568:
3215:
4292:. In English, the disjunctive word "or" is often understood exclusively, particularly when used with the particle "either". The English example below would normally be understood in conversation as implying that Mary is not both a singer and a poet.
2456:
4104:
7425:) and writing it to another drive. Under this method, if any one of the three hard drives are lost, the lost byte can be re-created by XORing bytes from the remaining drives. For instance, if the drive containing
3581:
5981:
7359:
On some computer architectures, it is more efficient to store a zero in a register by XOR-ing the register with itself (bits XOR-ed with themselves are always zero) than to load and store the value zero.
270:
342:
1626:
4564:
7330:
5122:
for exclusive disjunction (it seems that the mistake spreads widely), while neither in 1929 nor in other works did Łukasiewicz make such use. In fact, in 1949 Bocheński introduced a system of
7269:
7166:
3045:
5326:
5357:
4862:
969:
2493:
1396:
911:
824:
8539:
3140:
1036:
697:
2278:
1967:
4698:
4671:
4524:
3037:
2485:
2247:
1940:
1355:
995:
5170:
as exclusive disjunction has no relationship with the Polish "alternatywa rozłączna" of "exclusive or" and is an accident for which see the table on page 16 of the book in 1949.
671:
7520:
6277:
6154:
3852:
3749:
2013:
1683:
1097:
8568:
7039:
6946:
6503:
6247:
6104:
5835:
5805:
5627:
5485:
5426:
4274:
4096:
4067:
3962:
2384:
3793:
1178:
6328:
4497:
is almost fixedly used nowadays) and exclusive disjunction, and may also bring about confusions with its other uses, some classical and modern textbooks still keep such use.
3995:
1913:
943:
783:
731:
611:
8644:
6661:
6602:
6473:
6350:
5882:
5776:
5704:
5572:
1508:
1456:
1062:
8669:
8444:
6791:
6732:
6532:
6409:
5032:
2033:
1715:. With two inputs, XOR is true if and only if the inputs differ (one is true, one is false). With multiple inputs, XOR is true if and only if the number of true inputs is
1313:
8615:
8415:
6870:
6442:
6388:
5665:
5605:
5092:
850:
5295:
4955:
1482:
757:
8390:
7010:
6897:
6710:
6630:
6218:
6055:
5454:
5404:
2163:
2090:
1422:
645:
376:
6580:
582:
533:
507:
6760:
4977:
3931:
3899:
3879:
3674:
2328:
1993:
1287:
1235:
876:
8698:
4882:
4806:
4784:
4720:
2611:
2591:
2376:
1201:
1128:
191:
8735:
8510:
8473:
8344:
7557:
4930:
4906:
4495:
3698:
2631:
2352:
4345:
This behavior of
English "or" is also found in other languages. However, many languages have disjunctive constructions which are robustly exclusive such as French
8073:
Huntington, E. V. (1933). "New Sets of
Independent Postulates for the Algebra of Logic, With Special Reference to Whitehead and Russell's Principia Mathematica".
6974:
6924:
6182:
6132:
6082:
6028:
4455:
4429:
2304:
2223:
one is true and the other is false. For example, if two horses are racing, then one of the two will win the race, but not both of them. The exclusive disjunction
1261:
5728:
5533:
1151:
556:
5269:
5249:
5168:
5148:
5120:
5062:
5010:
4748:
4644:
4624:
4604:
4584:
4475:
4403:
4379:
4038:
4018:
2136:
2115:
1755:
4229:{\displaystyle {\begin{matrix}r=p\land q&\Leftrightarrow &r=p\cdot q{\pmod {2}}\\r=p\oplus q&\Leftrightarrow &r=p+q{\pmod {2}}\\\end{matrix}}}
4300:
However, disjunction can also be understood inclusively, even in combination with "either". For instance, the first example below shows that "either" can be
1633:
452:
2057:
E.g. row AB corresponds to the 2-circle, and row ABC to the 3-circle Venn diagram shown above. (As in the Venn diagrams, white is false, and red is true.)
7700:
4304:
used in combination with an outright statement that both disjuncts are true. The second example shows that the exclusive inference vanishes away under
4308:
contexts. If disjunction were understood as exclusive in this example, it would leave open the possibility that some people ate both rice and beans.
4477:
is a connective. Furthermore, Boole used it exclusively. Although such use does not show the relationship between inclusive disjunction (for which
7375:
systems. XOR is also heavily used in block ciphers such as AES (Rijndael) or
Serpent and in block cipher implementation (CBC, CFB, OFB or CTR).
1730:
that case. Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B".
8298:
4335:
7855:(2 ed.). San Diego, New York, Boston, London, Toronto, Sydney and Tokyo: A Harcourt Science and Technology Company. p. 51.
5912:
445:
223:
283:
3639:{\displaystyle \lnot (p\nleftrightarrow q)\Leftrightarrow \lnot p\nleftrightarrow q\Leftrightarrow p\nleftrightarrow \lnot q.}
7733:
8122:
7813:
Jennings quotes numerous authors saying that the word "or" has an exclusive sense. See
Chapter 3, "The First Myth of 'Or'":
1619:
111:
4533:
438:
83:
7216:
As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of two
7291:
3124:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&\lnot ((p\land q)\lor (\lnot p\land \lnot q))\end{matrix}}}
7226:
7123:
8291:
8271:
7762:
7574:
130:
4727:
90:
8272:
Proofs of XOR properties and applications of XOR, CS103: Mathematical
Foundations of Computing, Stanford University
7394:
2219:
Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is true
8150:
5300:
5331:
4786:
as non-equivalence literally which is possibly because it could be defined from negation and equivalence easily.
7942:
7609:
5197:
4327:
2573:
This representation of XOR may be found useful when constructing a circuit or network, because it has only one
2563:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)\lor (\lnot p\land q)\end{matrix}}}
68:
4815:
3210:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land (\lnot p\lor \lnot q)\end{matrix}}}
97:
5221:
948:
64:
1368:
881:
796:
8776:
8524:
8284:
8172:
1609:
1008:
676:
17:
8049:
4912:." Note that the Latin word "aut" means "exclusive or" and "vel" means "inclusive or", and that Peano use
2252:
1945:
5213:
5205:
4676:
4649:
4502:
3016:
2464:
2451:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land \lnot (p\land q)\end{matrix}}}
2226:
1918:
1326:
974:
79:
5130:
of classical logic which is a compatible extension of the notation of Łukasiewicz in 1929, and in which
650:
8176:
7505:
6262:
6139:
5095:
4984:
4323:
3813:
3710:
1998:
1656:
1067:
8553:
7024:
6931:
6488:
6232:
6089:
5820:
5790:
5612:
5470:
5411:
4250:
4072:
4043:
3938:
7379:
6294:
4988:
4331:
3803:. This unfortunately prevents the combination of these two systems into larger structures, such as a
3754:
1156:
6307:
5906:
3968:
1889:
922:
762:
710:
587:
8629:
7925:
7789:
7604:
6833:
6646:
6587:
6458:
6335:
5849:
5743:
5671:
5539:
5201:
5189:
3704:) are very useful in logic systems, they fail a more generalizable structure in the following way:
1487:
1435:
1041:
8654:
8429:
6776:
6717:
6517:
6394:
5017:
4700:
has the meaning of exclusive disjunction since the article is titled as "On the
Algebra of Logic".
2018:
1292:
8648:
8619:
8600:
8400:
7947:(in Russian) (3 ed.). МОСКВА: ГОСУДАРСТВЕННОЕ ИЗДАТЕЛЬСТВО ФИЗИКа-МАТЕМАТИЧЕСКОЙ ЛИТЕРАТУРЫ.
6843:
6415:
6361:
5638:
5578:
5071:
4527:
3654:
1556:
829:
276:
216:
57:
5274:
4937:
1461:
736:
155:
8375:
7882:
7624:
6989:
6876:
6689:
6609:
6197:
6034:
5433:
5383:
4755:
2142:
2069:
1734:
1526:
1401:
624:
355:
8108:
7970:
7750:
6559:
1704:
561:
512:
486:
8543:
8489:
7966:
7594:
7117:
7057:
6739:
5898:
5035:
4962:
4723:
4301:
4277:
3904:
3884:
3864:
3659:
3228:
2313:
1978:
1712:
1266:
1214:
855:
617:
417:
8683:
7837:
The
Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning
7713:
7281:
It is an optional bit-flipper (the deciding input chooses whether to invert the data input).
5271:, which may be interpreted as their elementwise exclusive or, has variously been denoted as
4867:
4791:
4769:
4705:
2596:
2576:
2361:
1183:
1110:
170:
8771:
8720:
8495:
8458:
8419:
8329:
7579:
7570:
7285:
5502:
5370:
5228:
5181:
4915:
4891:
4480:
3683:
3232:
2616:
2337:
1716:
1584:
703:
348:
35:
104:
8:
8673:
8448:
8196:
8114:
7835:
7589:
7383:
7349:
7113:
6953:
6903:
6161:
6111:
6061:
6007:
5984:
4434:
4408:
3800:
3701:
3677:
3575:
3221:
2307:
2283:
1594:
1240:
1207:
479:
7714:
5710:
5515:
1133:
538:
8781:
8308:
8003:
7479:
7464:
7454:
7209:
7068:
6808:
5254:
5234:
5153:
5133:
5127:
5105:
5047:
4995:
4733:
4629:
4609:
4589:
4569:
4460:
4388:
4364:
4305:
4023:
4003:
3804:
3236:
2121:
2100:
1740:
1708:
1589:
468:
407:
8138:
8034:
7356:
to add the numbers, and a series of AND, OR and NOT gates to create the carry output.
5099:
8752:
8572:
8319:
8193:
8118:
8021:
Huntington, E. V. (1904). "Sets of
Independent Postulates for the Algebra of Logic".
7758:
7729:
7483:
7471:
7096:
5905:
The exclusive or does not distribute over any binary function (not even itself), but
5185:
2052:
1531:
382:
8353:
8007:
4334:
and do not arise in downward entailing contexts if their calculation depends on the
1722:
It gains the name "exclusive or" because the meaning of "or" is ambiguous when both
8394:
8030:
7993:
7985:
7721:
7675:
5983:(Conjunction and exclusive or form the multiplication and addition operations of a
4957:
was used by Izrail
Solomonovich Gradshtein (Израиль Соломонович Градштейн) in 1936.
4319:
Examples such as the above have motivated analyses of the exclusivity inference as
4289:
4240:
1858:
1832:
1801:
1767:
1551:
1428:
7955:. Translated by Boddington, T. Oxford, London, New York and Paris: Pergamon Press.
7457:; however this is regarded as more of a curiosity and not encouraged in practice.
6836:
or self-inverse function; applying it twice leaves the variable input unchanged.
8164:
7644:
7475:
7460:
7418:
from two (or more) hard drives by XORing the just mentioned bytes, resulting in (
7341:
6817:
5123:
5065:
3997:
and has the added benefit of the arsenal of algebraic analysis tools for fields.
1536:
7989:
7840:. Cambridge/London: Macmillan, Barclay, & Macmillan/George Bell. p. 17.
2644:
8576:
8365:
8168:
7372:
7333:
7101:
6832:
Exclusive or with one specified input, as a function of the other input, is an
4980:
4809:
4763:
3240:
2220:
1758:
1546:
789:
7870:(3 ed.). New York, Dordrecht, Heidelberg and London: Springer. p. 3.
7725:
7367:, XOR is sometimes used as a simple, self-inverse mixing function, such as in
8765:
8714:
8710:
8323:
8276:
7918:
Notations de logique mathématique. Introduction au formulaire de mathématique
7619:
7614:
5039:
4338:. However, some researchers have treated exclusivity as a bona fide semantic
3858:
1561:
7404:
3–6 for creating parity information. For example, RAID can "back up" bytes
8266:
7720:. Translated by Bird, O. Dordrecht, Holland: D. Reidel Publishing Company.
7599:
7390:
7368:
7364:
7221:
5150:
for exclusive disjunction appeared at the first time. Bocheński's usage of
4759:
4751:
4382:
4040:
with 1, one can interpret the logical "AND" operation as multiplication on
2048:
1650:
8623:
8590:
8243:
7785:
7382:, modeling the XOR function requires a second layer because XOR is not a
5994:
2331:
2063:
1541:
1001:
163:
7998:
7665:
8423:
8218:
7634:
7337:
7084:
4339:
4320:
4244:
2043:
1579:
197:
7453:
XOR can be used to swap two numeric variables in computers, using the
7172:
Exclusive disjunction is often used for bitwise operations. Examples:
8547:
8369:
8201:
7796:(Winter 2016 ed.). Metaphysics Research Lab, Stanford University
7670:
1319:
7220:-bit strings is identical to the standard vector of addition in the
7079:
3257:
46:
8677:
8594:
8518:
8485:
7639:
7474:, XOR-based drawing methods are often used to manage such items as
7353:
7065:
4983:
in 1938. Shannon borrowed the symbol as exclusive disjunction from
2355:
1103:
31:
7902:
Vorlesungen über die Algebra der Logik (Exakte Logik), Erster Band
3246:
In summary, we have, in mathematical and in engineering notation:
8452:
7629:
7584:
6537:
6509:
6479:
6449:
5887:
5841:
5811:
5781:
5735:
4754:
in 1847, during the 40 years after Boole, his followers, such as
3965:. This field can represent any logic obtainable with the system
1723:
1646:
7336:
an odd number of the variables are true), which is equal to the
7708:(in French). The Netherlands: F. G. Kroonder, Bussum, Pays-Bas.
7105:
3796:
7978:
Transactions of the American Institute of Electrical Engineers
5976:{\displaystyle C\land (A\oplus B)=(C\land A)\oplus (C\land B)}
4109:
2214:
8191:
7499:
7274:
In computer science, exclusive disjunction has several uses:
5987:
5193:
5177:
3934:
1870:
1844:
1821:
1787:
265:{\displaystyle {\overline {x}}\cdot y+x\cdot {\overline {y}}}
7953:
Direct and Converse Theorems: The Elements of Symbolic Logic
7463:
leverage XOR properties in order to save space to represent
337:{\displaystyle ({\overline {x}}+{\overline {y}})\cdot (x+y)}
7887:
Studies in Logic by Members of the Johns Hopkins University
7401:
7045:
7015:
6981:
6797:
6767:
6667:
6637:
5491:
5461:
5209:
1807:
1773:
203:
8219:"Exclusive OR (XOR) and hardware random number generators"
7522:). Apart from the ASCII codes, the operator is encoded at
4342:
and proposed nonclassical logics which would validate it.
6283:
6253:
6223:
6189:
5217:
3227:
The exclusive or is also equivalent to the negation of a
3235:
is equivalent to the disjunction of the negation of its
2092:
shows that it outputs true whenever the inputs differ:
7889:. Boston: Little, Brown & Company. pp. 17–71.
5990:, and as in any field they obey the distributive law.)
3145:
3050:
2498:
2389:
8723:
8686:
8657:
8632:
8603:
8556:
8527:
8498:
8461:
8432:
8403:
8378:
8332:
8096:. New Jersey: Princeton University Press. p. 37.
7971:"A Symbolic Analysis of Relay and Switching Circuits"
7904:(in German). Leipzig: Druck und Verlag B. G. Teubner.
7508:
7294:
7229:
7126:
7027:
6992:
6956:
6934:
6906:
6879:
6846:
6779:
6742:
6720:
6692:
6649:
6612:
6590:
6562:
6520:
6491:
6461:
6418:
6397:
6364:
6338:
6310:
6265:
6235:
6200:
6164:
6142:
6114:
6092:
6064:
6037:
6010:
5915:
5852:
5823:
5793:
5746:
5713:
5674:
5641:
5615:
5581:
5542:
5518:
5473:
5436:
5414:
5386:
5334:
5303:
5277:
5257:
5237:
5156:
5136:
5108:
5074:
5050:
5020:
4998:
4965:
4940:
4918:
4894:
4870:
4818:
4794:
4772:
4736:
4708:
4679:
4652:
4632:
4612:
4592:
4572:
4536:
4505:
4483:
4463:
4437:
4411:
4391:
4367:
4253:
4107:
4075:
4046:
4026:
4006:
3971:
3941:
3907:
3887:
3867:
3816:
3757:
3713:
3686:
3662:
3584:
3255:
3143:
3048:
3019:
2642:
2633:
operations. A proof of this identity is given below:
2619:
2599:
2579:
2496:
2467:
2387:
2364:
2340:
2316:
2286:
2255:
2229:
2145:
2124:
2103:
2072:
2021:
2001:
1981:
1948:
1921:
1892:
1867:
1864:
1841:
1838:
1813:
1810:
1804:
1779:
1776:
1770:
1743:
1659:
1490:
1464:
1438:
1404:
1371:
1329:
1295:
1269:
1243:
1217:
1186:
1159:
1136:
1113:
1070:
1044:
1011:
977:
951:
925:
884:
858:
832:
799:
765:
739:
713:
679:
653:
627:
590:
564:
541:
515:
489:
358:
286:
226:
173:
7120:
representation. This is also the vector addition in
4559:{\displaystyle A\operatorname {\overline {\vee }} B}
4405:
mainly on classes, he also considered the case that
1847:
1790:
6552:When all inputs are true, the output is not true.
4283:
1861:
1835:
1818:
1784:
71:. Unsourced material may be challenged and removed.
8729:
8692:
8663:
8638:
8609:
8562:
8533:
8504:
8467:
8438:
8409:
8384:
8338:
7944:ПРЯМАЯ И ОБРАТНАЯ ТЕОРЕМЫ: ЭЛЕМЕНТЫ АЛГЕБРЫ ЛОГИКИ
7751:"9.2: Algebraic normal forms of Boolean functions"
7514:
7324:
7263:
7160:
7033:
7004:
6968:
6940:
6918:
6891:
6864:
6785:
6754:
6726:
6704:
6655:
6624:
6596:
6574:
6526:
6497:
6467:
6436:
6403:
6382:
6344:
6322:
6271:
6241:
6212:
6176:
6148:
6126:
6098:
6076:
6049:
6022:
5975:
5876:
5829:
5799:
5770:
5722:
5698:
5659:
5621:
5599:
5566:
5527:
5479:
5448:
5420:
5398:
5351:
5320:
5289:
5263:
5243:
5162:
5142:
5114:
5086:
5056:
5026:
5004:
4971:
4949:
4924:
4900:
4876:
4856:
4800:
4778:
4742:
4714:
4692:
4665:
4638:
4618:
4598:
4578:
4558:
4518:
4489:
4469:
4449:
4423:
4397:
4373:
4268:
4228:
4090:
4061:
4032:
4012:
3989:
3956:
3925:
3893:
3873:
3846:
3787:
3743:
3692:
3668:
3638:
3558:
3209:
3123:
3031:
3002:
2625:
2605:
2585:
2562:
2479:
2450:
2370:
2346:
2322:
2298:
2272:
2241:
2157:
2130:
2109:
2084:
2027:
2007:
1987:
1961:
1934:
1907:
1749:
1677:
1502:
1476:
1450:
1416:
1390:
1349:
1307:
1281:
1255:
1229:
1195:
1172:
1145:
1122:
1091:
1056:
1030:
989:
963:
937:
905:
870:
844:
818:
777:
751:
725:
691:
665:
639:
605:
576:
550:
527:
501:
370:
336:
264:
185:
8075:Transactions of the American Mathematical Society
8023:Transactions of the American Mathematical Society
7325:{\displaystyle A\oplus B\oplus C\oplus D\oplus E}
6682:When all inputs are false, the output is false.
5907:logical conjunction distributes over exclusive or
8763:
8163:
8149:] (in Polish) (1 ed.). Warsaw, Poland:
7663:
7264:{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{n}}
7161:{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{4}}
3220:This equivalence can be established by applying
8131:
8048:Leibniz, G. W. (1890) . Gerhardt, C. I. (ed.).
7694:
7692:
4288:Disjunction is often understood exclusively in
8306:
8110:Routledge Encyclopedia of Philosophy, Volume 8
8066:
8014:
7934:
7083:Traditional symbolic representation of an XOR
30:"XOR" redirects here. For the logic gate, see
8292:
7930:. Roma: Edizioni Cremonese. pp. 123–176.
7859:
6806:
4987:in 1904. Huntington borrowed the symbol from
4312:2. Mary is either a singer or a poet or both.
4276:, using this basis, is called the function's
3648:
3224:twice to the fourth line of the above proof.
1627:
446:
27:True when either but not both inputs are true
7893:
7868:A Concise Introduction to Mathematical Logic
7689:
6676:
3920:
3908:
3832:
3820:
3773:
3761:
3729:
3717:
2487:can also be expressed in the following way:
8137:
8051:Die philosophischen Schriften, Siebter Band
7844:
7829:
7827:
5321:{\displaystyle S\mathop {\triangledown } T}
3569:
2215:Equivalences, elimination, and introduction
8299:
8285:
8100:
8085:
8072:
8054:(in German). Berlin: Weidmann. p. 237
8020:
7950:
7865:
7780:
7778:
7776:
7774:
7494:It is also called "not left-right arrow" (
5352:{\displaystyle S\mathop {\vartriangle } T}
3231:, by the rules of material implication (a
1634:
1620:
453:
439:
7997:
7940:
7909:
7711:
7698:
7389:Similarly, XOR can be used in generating
7247:
7234:
7144:
7131:
5098:in 1949. Somebody may mistake that it is
4844:
4840:
4256:
4078:
4049:
3944:
131:Learn how and when to remove this message
7899:
7850:
7824:
7815:
7208:(this is equivalent to addition without
7100:
7078:
7060:values for true (1) and false (0), then
4857:{\displaystyle a\circ b=a-b\,\cup \,b-a}
4326:calculated on the basis of an inclusive
2055:XOR of the arguments shown on the left.
2042:
1645:
8047:
7965:
7874:
7794:The Stanford Encyclopedia of Philosophy
7771:
7446:can be XORed to recover the lost byte.
6546:
4069:and the "XOR" operation as addition on
3810:However, the system using exclusive or
964:{\displaystyle A\not \Leftrightarrow B}
14:
8764:
8216:
8091:
7657:
7278:It tells whether two bits are unequal.
5188:exclusive or operator, beginning with
4750:as equivalence could be dated back to
4530:in 1883. Strictly speaking, Ladd used
4352:
1391:{\displaystyle A{\underline {\lor }}B}
906:{\displaystyle {\overline {A\cdot B}}}
819:{\displaystyle A{\overline {\land }}B}
8534:{\displaystyle \not \leftrightarrow }
8280:
8241:
8217:Davies, Robert B (28 February 2002).
8192:
8106:
7923:
7915:
7833:
7784:
7664:Germundsson, Roger; Weisstein, Eric.
5500:
5368:
4000:More specifically, if one associates
1031:{\displaystyle A{\overline {\lor }}B}
692:{\displaystyle A\leftrightharpoons B}
7906:Reprinted by Thoemmes Press in 2000.
7880:
7853:A Mathematical Introduction to Logic
7820:. New York: Oxford University Press.
7748:
7090:
4992:(logical sum) too, and in 1933 used
2273:{\displaystyle p\operatorname {?} q}
1962:{\displaystyle {\underline {\vee }}}
69:adding citations to reliable sources
40:
7074:
6814:
6292:
4693:{\displaystyle {\overline {\vee }}}
4666:{\displaystyle {\overline {\vee }}}
4519:{\displaystyle {\overline {\vee }}}
4315:3. Nobody ate either rice or beans.
4214:
4207:
4157:
4150:
3032:{\displaystyle p\nleftrightarrow q}
2480:{\displaystyle p\nleftrightarrow q}
2306:, can be expressed in terms of the
2242:{\displaystyle p\nleftrightarrow q}
1935:{\displaystyle {\overline {\vee }}}
1350:{\displaystyle A\ {\text{XNOR}}\ B}
990:{\displaystyle A\nleftrightarrow B}
24:
8724:
8499:
8333:
8094:Introduction to Mathematical Logic
5992:
5896:
3627:
3606:
3585:
3491:
3399:
3390:
3304:
3285:
3194:
3185:
3105:
3096:
3069:
2956:
2915:
2906:
2875:
2832:
2823:
2808:
2768:
2740:
2728:
2691:
2672:
2580:
2541:
2526:
2426:
2365:
1114:
666:{\displaystyle A\Leftrightarrow B}
597:
594:
568:
25:
8793:
8260:
8035:10.1090/S0002-9947-1904-1500675-4
7515:{\displaystyle \nleftrightarrow }
7395:hardware random number generators
6272:{\displaystyle \nLeftrightarrow }
6149:{\displaystyle \nLeftrightarrow }
3847:{\displaystyle (\{T,F\},\oplus )}
3744:{\displaystyle (\{T,F\},\wedge )}
2008:{\displaystyle \nleftrightarrow }
1678:{\displaystyle A\oplus B\oplus C}
1092:{\displaystyle {\overline {A+B}}}
8746:
8563:{\displaystyle \leftrightarrow }
8352:
8181:. Prentice-Hall. pp. 44–46.
8173:"2.9: Bitwise logical operators"
7044:
7034:{\displaystyle \Leftrightarrow }
7014:
6980:
6941:{\displaystyle \Leftrightarrow }
6796:
6766:
6666:
6636:
6536:
6508:
6498:{\displaystyle \Leftrightarrow }
6478:
6448:
6282:
6252:
6242:{\displaystyle \Leftrightarrow }
6222:
6188:
6099:{\displaystyle \Leftrightarrow }
5886:
5840:
5830:{\displaystyle \Leftrightarrow }
5810:
5800:{\displaystyle \Leftrightarrow }
5780:
5734:
5622:{\displaystyle \Leftrightarrow }
5490:
5480:{\displaystyle \Leftrightarrow }
5460:
5421:{\displaystyle \Leftrightarrow }
5034:, also denoting the negation of
4673:as exclusions, while implicitly
4284:Exclusive or in natural language
4269:{\displaystyle \mathbb {F} _{2}}
4091:{\displaystyle \mathbb {F} _{2}}
4062:{\displaystyle \mathbb {F} _{2}}
3957:{\displaystyle \mathbb {F} _{2}}
3861:. The combination of operators
3013:It is sometimes useful to write
1857:
1831:
1800:
1766:
1605:
1604:
202:
154:
45:
8242:Nobel, Rickard (26 July 2011).
8235:
8210:
8185:
8157:
8041:
7959:
7757:. CRC Press. pp. 285–286.
4730:in 1890, Although the usage of
3788:{\displaystyle (\{T,F\},\lor )}
1173:{\displaystyle {\overline {A}}}
56:needs additional citations for
8604:
8557:
8433:
8404:
8379:
8147:Elements of Mathematical Logic
7807:
7742:
7716:A Precis of Mathematical Logic
7702:Précis de logique mathématique
7610:List of Boolean algebra topics
7378:In simple threshold-activated
7348:In logical circuits, a simple
7252:
7230:
7149:
7127:
7028:
6935:
6826:
6780:
6721:
6521:
6492:
6431:
6419:
6398:
6377:
6365:
6323:{\displaystyle A\rightarrow B}
6314:
6236:
6093:
5970:
5958:
5952:
5940:
5934:
5922:
5824:
5794:
5654:
5642:
5616:
5594:
5582:
5474:
5415:
4296:1. Mary is a singer or a poet.
4218:
4208:
4186:
4161:
4151:
4129:
3990:{\displaystyle (\land ,\lor )}
3984:
3972:
3841:
3817:
3782:
3758:
3738:
3714:
3618:
3603:
3600:
3588:
3549:
3531:
3528:
3516:
3506:
3494:
3481:
3469:
3456:
3430:
3427:
3415:
3405:
3387:
3377:
3365:
3316:
3301:
3291:
3276:
3200:
3182:
3176:
3164:
3114:
3111:
3093:
3087:
3075:
3072:
2993:
2981:
2971:
2959:
2943:
2931:
2921:
2903:
2890:
2887:
2872:
2866:
2854:
2851:
2841:
2838:
2820:
2814:
2799:
2796:
2783:
2774:
2759:
2756:
2746:
2734:
2719:
2716:
2703:
2688:
2678:
2663:
2593:operation and small number of
2553:
2538:
2532:
2517:
2441:
2429:
2420:
2408:
1908:{\displaystyle {\dot {\vee }}}
1494:
1442:
1048:
938:{\displaystyle A\not \equiv B}
836:
778:{\displaystyle A\rightarrow B}
769:
726:{\displaystyle A\Rightarrow B}
717:
683:
657:
606:{\displaystyle A\&\&B}
331:
319:
313:
287:
180:
174:
13:
1:
8639:{\displaystyle \nrightarrow }
8151:Państwowe Wydawnictwo Naukowe
8143:Elementy logiki matematycznej
7792:. In Zalta, Edward N. (ed.).
7554:⊕, ⊕
7284:It tells whether there is an
6656:{\displaystyle \nRightarrow }
6597:{\displaystyle \nRightarrow }
6468:{\displaystyle \nRightarrow }
6345:{\displaystyle \nRightarrow }
5877:{\displaystyle ~~~\oplus ~~~}
5771:{\displaystyle ~~~\oplus ~~~}
5699:{\displaystyle ~~~\oplus ~~~}
5567:{\displaystyle ~~~\oplus ~~~}
5363:
4385:in 1847. Although Boole used
4330:. Implicatures are typically
2038:
1503:{\displaystyle A\leftarrow B}
1451:{\displaystyle A\Leftarrow B}
1057:{\displaystyle A\downarrow B}
8664:{\displaystyle \nleftarrow }
8439:{\displaystyle \rightarrow }
7818:The Genealogy of Disjunction
7489:
6786:{\displaystyle \Rightarrow }
6727:{\displaystyle \Rightarrow }
6527:{\displaystyle \rightarrow }
6404:{\displaystyle \rightarrow }
5027:{\displaystyle \not \equiv }
4685:
4658:
4545:
4511:
3544:
3451:
3438:
3347:
3334:
2028:{\displaystyle \not \equiv }
1927:
1308:{\displaystyle A\parallel B}
1165:
1084:
1020:
898:
808:
308:
295:
257:
232:
7:
8610:{\displaystyle \downarrow }
8410:{\displaystyle \leftarrow }
8244:"How RAID 5 actually works"
7990:10.1109/T-AIEE.1938.5057767
7563:
6865:{\displaystyle ~A\oplus B~}
6437:{\displaystyle (B\oplus C)}
6383:{\displaystyle (A\oplus C)}
5660:{\displaystyle (A\oplus B)}
5600:{\displaystyle (B\oplus C)}
5180:, has been used in several
5087:{\displaystyle J\phi \psi }
4722:, denoting the negation of
4324:conversational implicatures
845:{\displaystyle A\uparrow B}
10:
8798:
8178:The C Programming Language
7951:Gradshtein, I. S. (1963).
7941:ГРАДШТЕЙН, И. С. (1959) .
7885:. In Peirce, C. S. (ed.).
7380:artificial neural networks
7094:
5290:{\displaystyle S\ominus T}
4985:Edward Vermilye Huntington
4950:{\displaystyle \vee \vee }
3649:Relation to modern algebra
3574:By applying the spirit of
2461:The exclusive disjunction
2051:is the truth table of the
1477:{\displaystyle A\subset B}
752:{\displaystyle A\supset B}
29:
8743:
8706:
8586:
8481:
8385:{\displaystyle \uparrow }
8361:
8350:
8315:
7920:. Turin: Fratelli Boccna.
7883:"On the Algebra of Logic"
7755:Algorithmic Cryptanalysis
7726:10.1007/978-94-017-0592-9
7712:Bocheński, J. M. (1959).
7699:Bocheński, J. M. (1949).
7005:{\displaystyle ~\oplus ~}
6892:{\displaystyle ~\oplus ~}
6705:{\displaystyle A\oplus B}
6678:Falsehood-preserving: yes
6625:{\displaystyle A\oplus B}
6213:{\displaystyle ~\oplus ~}
6050:{\displaystyle ~\oplus ~}
5449:{\displaystyle B\oplus A}
5399:{\displaystyle A\oplus B}
5126:that names all 16 binary
5012:as inclusive disjunction.
4989:Gottfried Wilhelm Leibniz
4932:as inclusive disjunction.
3239:and its consequence) and
2158:{\displaystyle A\oplus B}
2085:{\displaystyle A\oplus B}
1417:{\displaystyle A\oplus B}
640:{\displaystyle A\equiv B}
434:
426:
416:
406:
398:
390:
381:
371:{\displaystyle x\oplus y}
347:
275:
215:
210:
196:
162:
153:
148:
7881:Ladd, Christine (1883).
7866:Rautenberg, W. (2010) .
7816:Jennings, R. E. (1994).
7651:
7041:
7021:
6948:
6928:
6793:
6773:
6734:
6714:
6663:
6643:
6604:
6584:
6575:{\displaystyle A\land B}
6505:
6485:
6475:
6455:
6352:
6332:
6279:
6259:
6249:
6229:
6156:
6136:
6106:
6086:
5837:
5817:
5807:
5787:
5629:
5609:
5487:
5467:
5428:
5408:
5102:who is the first to use
3570:Negation of the operator
2047:Each row of this binary
577:{\displaystyle A\&B}
528:{\displaystyle A\cdot B}
502:{\displaystyle A\land B}
8649:Converse nonimplication
7927:Opere Scelte, Volume II
7288:number of 1 bits (
6824:The function is linear.
6755:{\displaystyle A\lor B}
4972:{\displaystyle \oplus }
4766:and so on, did not use
4528:Christine Ladd-Franklin
3933:produce the well-known
3926:{\displaystyle \{T,F\}}
3894:{\displaystyle \oplus }
3874:{\displaystyle \wedge }
3669:{\displaystyle \wedge }
2323:{\displaystyle \wedge }
1988:{\displaystyle \oplus }
1737:by the prefix operator
1701:logical non-equivalence
1557:Functional completeness
1282:{\displaystyle A\mid B}
1230:{\displaystyle A\lor B}
871:{\displaystyle A\mid B}
8731:
8694:
8693:{\displaystyle \land }
8665:
8640:
8611:
8564:
8535:
8506:
8469:
8440:
8411:
8386:
8340:
8197:"Symmetric Difference"
8107:Craig, Edward (1998).
7851:Enderton, H. (2001) .
7749:Joux, Antoine (2009).
7625:Propositional calculus
7558:mathematical operators
7516:
7326:
7265:
7169:
7162:
7087:
7035:
7006:
6970:
6942:
6920:
6893:
6866:
6787:
6756:
6728:
6706:
6657:
6626:
6598:
6576:
6528:
6499:
6469:
6438:
6405:
6384:
6346:
6324:
6273:
6243:
6214:
6178:
6150:
6128:
6100:
6078:
6051:
6024:
5977:
5878:
5831:
5801:
5772:
5724:
5700:
5661:
5623:
5601:
5568:
5529:
5481:
5450:
5422:
5400:
5353:
5322:
5291:
5265:
5245:
5164:
5144:
5116:
5088:
5058:
5028:
5006:
4973:
4951:
4926:
4902:
4878:
4877:{\displaystyle \circ }
4858:
4802:
4801:{\displaystyle \circ }
4780:
4779:{\displaystyle \not =}
4756:Charles Sanders Peirce
4744:
4716:
4715:{\displaystyle \not =}
4694:
4667:
4640:
4620:
4600:
4580:
4560:
4520:
4491:
4471:
4451:
4425:
4399:
4375:
4270:
4230:
4092:
4063:
4034:
4014:
3991:
3958:
3927:
3895:
3875:
3848:
3789:
3745:
3694:
3670:
3640:
3560:
3211:
3125:
3039:in the following way:
3033:
3004:
2627:
2607:
2606:{\displaystyle \land }
2587:
2586:{\displaystyle \lnot }
2564:
2481:
2452:
2372:
2371:{\displaystyle \lnot }
2348:
2324:
2300:
2274:
2243:
2159:
2132:
2111:
2086:
2059:
2029:
2009:
1989:
1963:
1936:
1909:
1751:
1711:whose negation is the
1685:
1679:
1527:Propositional calculus
1504:
1478:
1452:
1418:
1392:
1351:
1309:
1283:
1257:
1231:
1197:
1196:{\displaystyle \sim A}
1174:
1147:
1124:
1123:{\displaystyle \neg A}
1093:
1058:
1032:
991:
965:
939:
907:
872:
846:
820:
779:
753:
727:
693:
667:
641:
607:
578:
552:
529:
503:
372:
338:
266:
187:
186:{\displaystyle (0110)}
34:. For other uses, see
8753:Philosophy portal
8732:
8730:{\displaystyle \bot }
8695:
8666:
8641:
8612:
8565:
8536:
8507:
8505:{\displaystyle \neg }
8470:
8468:{\displaystyle \lor }
8441:
8412:
8387:
8341:
8339:{\displaystyle \top }
7900:Schröder, E. (1890).
7595:Disjunctive syllogism
7517:
7327:
7266:
7163:
7104:
7082:
7036:
7007:
6971:
6943:
6921:
6894:
6867:
6788:
6757:
6729:
6707:
6658:
6627:
6599:
6577:
6529:
6500:
6470:
6439:
6406:
6385:
6347:
6325:
6274:
6244:
6215:
6179:
6151:
6129:
6101:
6079:
6052:
6025:
5978:
5879:
5832:
5802:
5773:
5725:
5701:
5662:
5624:
5602:
5569:
5530:
5482:
5451:
5423:
5401:
5354:
5323:
5292:
5266:
5246:
5182:programming languages
5165:
5145:
5117:
5096:Józef Maria Bocheński
5089:
5059:
5029:
5007:
4974:
4952:
4927:
4925:{\displaystyle \cup }
4903:
4901:{\displaystyle \cup }
4884:corresponds to Latin
4879:
4859:
4803:
4781:
4745:
4717:
4695:
4668:
4641:
4621:
4601:
4581:
4561:
4521:
4492:
4490:{\displaystyle \vee }
4472:
4452:
4426:
4400:
4376:
4278:algebraic normal form
4271:
4239:The description of a
4231:
4093:
4064:
4035:
4015:
3992:
3959:
3928:
3896:
3876:
3849:
3790:
3746:
3695:
3693:{\displaystyle \lor }
3671:
3641:
3561:
3229:logical biconditional
3212:
3126:
3034:
3005:
2628:
2626:{\displaystyle \lor }
2608:
2588:
2565:
2482:
2453:
2373:
2349:
2347:{\displaystyle \lor }
2325:
2301:
2275:
2244:
2160:
2133:
2112:
2087:
2046:
2030:
2010:
1990:
1964:
1937:
1910:
1752:
1713:logical biconditional
1697:exclusive alternation
1693:exclusive disjunction
1680:
1649:
1585:Programming languages
1505:
1479:
1453:
1419:
1393:
1352:
1310:
1284:
1258:
1232:
1198:
1175:
1148:
1125:
1094:
1059:
1033:
992:
966:
940:
908:
873:
847:
821:
780:
754:
728:
694:
668:
642:
608:
579:
553:
530:
504:
373:
339:
267:
188:
8721:
8684:
8655:
8630:
8601:
8554:
8525:
8496:
8459:
8430:
8401:
8395:Converse implication
8376:
8330:
8115:Taylor & Francis
8092:Church, A. (1996) .
7580:Affirming a disjunct
7571:Material conditional
7506:
7352:can be made with an
7292:
7227:
7124:
7114:nonnegative integers
7025:
6990:
6954:
6932:
6904:
6877:
6844:
6777:
6740:
6718:
6690:
6647:
6610:
6588:
6560:
6548:Truth-preserving: no
6518:
6489:
6459:
6416:
6395:
6362:
6336:
6308:
6263:
6233:
6198:
6162:
6140:
6112:
6090:
6062:
6035:
6008:
5913:
5850:
5821:
5791:
5744:
5711:
5672:
5639:
5613:
5579:
5540:
5516:
5471:
5434:
5412:
5384:
5332:
5301:
5275:
5255:
5235:
5229:symmetric difference
5154:
5134:
5106:
5072:
5048:
5018:
4996:
4963:
4938:
4916:
4892:
4868:
4816:
4792:
4770:
4734:
4706:
4677:
4650:
4630:
4610:
4590:
4570:
4534:
4503:
4481:
4461:
4435:
4431:are propositions in
4409:
4389:
4365:
4251:
4105:
4073:
4044:
4024:
4004:
3969:
3939:
3905:
3885:
3865:
3814:
3755:
3711:
3684:
3660:
3582:
3253:
3241:material equivalence
3233:material conditional
3141:
3046:
3017:
2640:
2617:
2597:
2577:
2494:
2465:
2385:
2362:
2338:
2314:
2284:
2253:
2227:
2143:
2122:
2101:
2070:
2019:
1999:
1979:
1946:
1919:
1890:
1741:
1657:
1488:
1462:
1436:
1402:
1369:
1327:
1293:
1267:
1241:
1215:
1184:
1157:
1134:
1111:
1068:
1042:
1009:
975:
949:
923:
882:
856:
830:
797:
763:
737:
711:
677:
651:
625:
588:
562:
539:
513:
487:
356:
349:Zhegalkin polynomial
284:
224:
171:
65:improve this article
36:XOR (disambiguation)
8777:Logical connectives
8309:logical connectives
8165:Kernighan, Brian W.
7590:Controlled NOT gate
7486:or overlay planes.
7482:on systems without
7064:works exactly like
6969:{\displaystyle ~A~}
6919:{\displaystyle ~B~}
6177:{\displaystyle ~A~}
6127:{\displaystyle ~0~}
6077:{\displaystyle ~A~}
6023:{\displaystyle ~A~}
5192:and also including
4450:{\displaystyle x+y}
4424:{\displaystyle x,y}
4353:Alternative symbols
3799:, but neither is a
2308:logical conjunction
2299:{\displaystyle Jpq}
1595:Philosophy of logic
1256:{\displaystyle A+B}
469:Logical connectives
145:
8727:
8690:
8661:
8636:
8607:
8560:
8531:
8502:
8465:
8436:
8407:
8382:
8366:Alternative denial
8336:
8194:Weisstein, Eric W.
8169:Ritchie, Dennis M.
7924:Peano, G. (1958).
7916:Peano, G. (1894).
7834:Boole, G. (1847).
7512:
7465:doubly linked list
7455:XOR swap algorithm
7384:linearly separable
7322:
7261:
7170:
7158:
7088:
7031:
7002:
6966:
6938:
6916:
6889:
6862:
6783:
6752:
6724:
6702:
6653:
6622:
6594:
6572:
6524:
6495:
6465:
6434:
6401:
6380:
6342:
6320:
6269:
6239:
6210:
6174:
6146:
6124:
6096:
6074:
6047:
6020:
5973:
5874:
5827:
5797:
5768:
5723:{\displaystyle ~C}
5720:
5696:
5657:
5619:
5597:
5564:
5528:{\displaystyle ~A}
5525:
5477:
5446:
5418:
5396:
5349:
5318:
5287:
5261:
5241:
5160:
5140:
5112:
5084:
5054:
5024:
5002:
4969:
4947:
4922:
4898:
4874:
4854:
4798:
4776:
4740:
4712:
4690:
4663:
4636:
4616:
4596:
4576:
4556:
4516:
4487:
4467:
4457:, and at the time
4447:
4421:
4395:
4371:
4306:downward entailing
4266:
4226:
4224:
4088:
4059:
4030:
4010:
3987:
3954:
3935:two-element field
3923:
3891:
3871:
3844:
3785:
3741:
3690:
3666:
3636:
3556:
3554:
3207:
3205:
3121:
3119:
3029:
3000:
2998:
2623:
2603:
2583:
2560:
2558:
2477:
2448:
2446:
2368:
2344:
2320:
2296:
2270:
2249:, also denoted by
2239:
2155:
2128:
2107:
2082:
2060:
2025:
2005:
1985:
1959:
1957:
1932:
1905:
1747:
1705:logical inequality
1686:
1675:
1590:Mathematical logic
1500:
1474:
1448:
1414:
1388:
1383:
1347:
1305:
1279:
1253:
1227:
1193:
1170:
1146:{\displaystyle -A}
1143:
1120:
1089:
1054:
1028:
987:
961:
935:
903:
868:
842:
816:
775:
749:
723:
689:
663:
637:
603:
574:
551:{\displaystyle AB}
548:
525:
499:
368:
334:
262:
183:
143:
8759:
8758:
7735:978-90-481-8329-6
7556:), both in block
7502:-based markdown (
7472:computer graphics
7467:data structures.
7097:Bitwise operation
7091:Bitwise operation
7052:
7051:
7001:
6995:
6965:
6959:
6915:
6909:
6888:
6882:
6861:
6849:
6804:
6803:
6674:
6673:
6544:
6543:
6290:
6289:
6209:
6203:
6173:
6167:
6123:
6117:
6073:
6067:
6046:
6040:
6019:
6013:
5894:
5893:
5873:
5870:
5867:
5861:
5858:
5855:
5767:
5764:
5761:
5755:
5752:
5749:
5716:
5695:
5692:
5689:
5683:
5680:
5677:
5563:
5560:
5557:
5551:
5548:
5545:
5521:
5498:
5497:
5264:{\displaystyle T}
5244:{\displaystyle S}
5163:{\displaystyle J}
5143:{\displaystyle J}
5115:{\displaystyle J}
5057:{\displaystyle J}
5005:{\displaystyle +}
4743:{\displaystyle =}
4688:
4661:
4639:{\displaystyle B}
4619:{\displaystyle A}
4599:{\displaystyle B}
4579:{\displaystyle A}
4548:
4514:
4470:{\displaystyle +}
4398:{\displaystyle +}
4374:{\displaystyle +}
4336:Maxim of Quantity
4290:natural languages
4033:{\displaystyle T}
4013:{\displaystyle F}
3805:mathematical ring
3547:
3454:
3441:
3350:
3337:
2212:
2211:
2131:{\displaystyle B}
2110:{\displaystyle A}
2058:
1950:
1930:
1902:
1750:{\displaystyle J}
1644:
1643:
1513:
1512:
1376:
1343:
1339:
1335:
1168:
1087:
1023:
901:
811:
463:
462:
311:
298:
260:
235:
141:
140:
133:
115:
16:(Redirected from
8789:
8751:
8750:
8749:
8736:
8734:
8733:
8728:
8699:
8697:
8696:
8691:
8670:
8668:
8667:
8662:
8645:
8643:
8642:
8637:
8616:
8614:
8613:
8608:
8569:
8567:
8566:
8561:
8540:
8538:
8537:
8532:
8511:
8509:
8508:
8503:
8474:
8472:
8471:
8466:
8445:
8443:
8442:
8437:
8416:
8414:
8413:
8408:
8391:
8389:
8388:
8383:
8356:
8345:
8343:
8342:
8337:
8301:
8294:
8287:
8278:
8277:
8255:
8254:
8252:
8250:
8239:
8233:
8232:
8230:
8228:
8223:
8214:
8208:
8207:
8206:
8189:
8183:
8182:
8161:
8155:
8154:
8139:Łukasiewicz, Jan
8135:
8129:
8128:
8124:978-0-41507310-3
8104:
8098:
8097:
8089:
8083:
8082:
8070:
8064:
8063:
8061:
8059:
8045:
8039:
8038:
8018:
8012:
8011:
8001:
7975:
7963:
7957:
7956:
7948:
7938:
7932:
7931:
7921:
7913:
7907:
7905:
7897:
7891:
7890:
7878:
7872:
7871:
7863:
7857:
7856:
7848:
7842:
7841:
7831:
7822:
7821:
7811:
7805:
7804:
7802:
7801:
7782:
7769:
7768:
7746:
7740:
7739:
7719:
7709:
7707:
7696:
7687:
7686:
7684:
7682:
7676:Wolfram Research
7661:
7555:
7551:
7548:
7545:
7543:
7538:
7534:
7531:
7528:
7526:
7521:
7519:
7518:
7513:
7497:
7496:\nleftrightarrow
7461:XOR linked lists
7445:
7444:
7438:
7437:
7431:
7430:
7424:
7423:
7417:
7416:
7410:
7409:
7331:
7329:
7328:
7323:
7270:
7268:
7267:
7262:
7260:
7259:
7250:
7242:
7237:
7207:
7206:
7200:
7199:
7193:
7192:
7167:
7165:
7164:
7159:
7157:
7156:
7147:
7139:
7134:
7108:addition is the
7075:Computer science
7048:
7040:
7038:
7037:
7032:
7018:
7011:
7009:
7008:
7003:
6999:
6993:
6984:
6975:
6973:
6972:
6967:
6963:
6957:
6947:
6945:
6944:
6939:
6925:
6923:
6922:
6917:
6913:
6907:
6898:
6896:
6895:
6890:
6886:
6880:
6871:
6869:
6868:
6863:
6859:
6847:
6838:
6837:
6800:
6792:
6790:
6789:
6784:
6770:
6761:
6759:
6758:
6753:
6733:
6731:
6730:
6725:
6711:
6709:
6708:
6703:
6684:
6683:
6670:
6662:
6660:
6659:
6654:
6640:
6631:
6629:
6628:
6623:
6603:
6601:
6600:
6595:
6581:
6579:
6578:
6573:
6554:
6553:
6540:
6533:
6531:
6530:
6525:
6512:
6504:
6502:
6501:
6496:
6482:
6474:
6472:
6471:
6466:
6452:
6443:
6441:
6440:
6435:
6410:
6408:
6407:
6402:
6389:
6387:
6386:
6381:
6351:
6349:
6348:
6343:
6329:
6327:
6326:
6321:
6302:
6301:
6286:
6278:
6276:
6275:
6270:
6256:
6248:
6246:
6245:
6240:
6226:
6219:
6217:
6216:
6211:
6207:
6201:
6192:
6183:
6181:
6180:
6175:
6171:
6165:
6155:
6153:
6152:
6147:
6133:
6131:
6130:
6125:
6121:
6115:
6105:
6103:
6102:
6097:
6083:
6081:
6080:
6075:
6071:
6065:
6056:
6054:
6053:
6048:
6044:
6038:
6029:
6027:
6026:
6021:
6017:
6011:
6002:
6001:
5982:
5980:
5979:
5974:
5890:
5883:
5881:
5880:
5875:
5871:
5868:
5865:
5859:
5856:
5853:
5844:
5836:
5834:
5833:
5828:
5814:
5806:
5804:
5803:
5798:
5784:
5777:
5775:
5774:
5769:
5765:
5762:
5759:
5753:
5750:
5747:
5738:
5729:
5727:
5726:
5721:
5714:
5705:
5703:
5702:
5697:
5693:
5690:
5687:
5681:
5678:
5675:
5666:
5664:
5663:
5658:
5628:
5626:
5625:
5620:
5606:
5604:
5603:
5598:
5573:
5571:
5570:
5565:
5561:
5558:
5555:
5549:
5546:
5543:
5534:
5532:
5531:
5526:
5519:
5510:
5509:
5494:
5486:
5484:
5483:
5478:
5464:
5455:
5453:
5452:
5447:
5427:
5425:
5424:
5419:
5405:
5403:
5402:
5397:
5378:
5377:
5358:
5356:
5355:
5350:
5342:
5327:
5325:
5324:
5319:
5311:
5296:
5294:
5293:
5288:
5270:
5268:
5267:
5262:
5250:
5248:
5247:
5242:
5175:
5169:
5167:
5166:
5161:
5149:
5147:
5146:
5141:
5121:
5119:
5118:
5113:
5093:
5091:
5090:
5085:
5063:
5061:
5060:
5055:
5033:
5031:
5030:
5025:
5011:
5009:
5008:
5003:
4978:
4976:
4975:
4970:
4956:
4954:
4953:
4948:
4931:
4929:
4928:
4923:
4907:
4905:
4904:
4899:
4883:
4881:
4880:
4875:
4863:
4861:
4860:
4855:
4807:
4805:
4804:
4799:
4785:
4783:
4782:
4777:
4749:
4747:
4746:
4741:
4721:
4719:
4718:
4713:
4699:
4697:
4696:
4691:
4689:
4681:
4672:
4670:
4669:
4664:
4662:
4654:
4645:
4643:
4642:
4637:
4625:
4623:
4622:
4617:
4605:
4603:
4602:
4597:
4585:
4583:
4582:
4577:
4565:
4563:
4562:
4557:
4549:
4541:
4525:
4523:
4522:
4517:
4515:
4507:
4496:
4494:
4493:
4488:
4476:
4474:
4473:
4468:
4456:
4454:
4453:
4448:
4430:
4428:
4427:
4422:
4404:
4402:
4401:
4396:
4380:
4378:
4377:
4372:
4275:
4273:
4272:
4267:
4265:
4264:
4259:
4241:Boolean function
4235:
4233:
4232:
4227:
4225:
4221:
4164:
4097:
4095:
4094:
4089:
4087:
4086:
4081:
4068:
4066:
4065:
4060:
4058:
4057:
4052:
4039:
4037:
4036:
4031:
4019:
4017:
4016:
4011:
3996:
3994:
3993:
3988:
3963:
3961:
3960:
3955:
3953:
3952:
3947:
3932:
3930:
3929:
3924:
3900:
3898:
3897:
3892:
3880:
3878:
3877:
3872:
3853:
3851:
3850:
3845:
3794:
3792:
3791:
3786:
3750:
3748:
3747:
3742:
3699:
3697:
3696:
3691:
3675:
3673:
3672:
3667:
3645:
3643:
3642:
3637:
3576:De Morgan's laws
3565:
3563:
3562:
3557:
3555:
3548:
3543:
3535:
3462:
3455:
3447:
3442:
3434:
3358:
3351:
3343:
3338:
3330:
3222:De Morgan's laws
3216:
3214:
3213:
3208:
3206:
3130:
3128:
3127:
3122:
3120:
3038:
3036:
3035:
3030:
3009:
3007:
3006:
3001:
2999:
2949:
2896:
2789:
2709:
2632:
2630:
2629:
2624:
2612:
2610:
2609:
2604:
2592:
2590:
2589:
2584:
2569:
2567:
2566:
2561:
2559:
2486:
2484:
2483:
2478:
2457:
2455:
2454:
2449:
2447:
2377:
2375:
2374:
2369:
2353:
2351:
2350:
2345:
2329:
2327:
2326:
2321:
2310:("logical and",
2305:
2303:
2302:
2297:
2279:
2277:
2276:
2271:
2263:
2248:
2246:
2245:
2240:
2164:
2162:
2161:
2156:
2137:
2135:
2134:
2129:
2116:
2114:
2113:
2108:
2095:
2094:
2091:
2089:
2088:
2083:
2056:
2034:
2032:
2031:
2026:
2014:
2012:
2011:
2006:
1994:
1992:
1991:
1986:
1973:
1968:
1966:
1965:
1960:
1958:
1941:
1939:
1938:
1933:
1931:
1923:
1914:
1912:
1911:
1906:
1904:
1903:
1895:
1877:
1876:
1873:
1872:
1869:
1866:
1863:
1854:
1853:
1850:
1849:
1846:
1843:
1840:
1837:
1828:
1827:
1824:
1823:
1820:
1816:
1815:
1812:
1809:
1806:
1797:
1796:
1793:
1792:
1789:
1786:
1782:
1781:
1778:
1775:
1772:
1756:
1754:
1753:
1748:
1709:logical operator
1684:
1682:
1681:
1676:
1636:
1629:
1622:
1608:
1607:
1552:Boolean function
1518:Related concepts
1509:
1507:
1506:
1501:
1483:
1481:
1480:
1475:
1457:
1455:
1454:
1449:
1423:
1421:
1420:
1415:
1397:
1395:
1394:
1389:
1384:
1356:
1354:
1353:
1348:
1341:
1340:
1337:
1333:
1314:
1312:
1311:
1306:
1288:
1286:
1285:
1280:
1262:
1260:
1259:
1254:
1236:
1234:
1233:
1228:
1202:
1200:
1199:
1194:
1179:
1177:
1176:
1171:
1169:
1161:
1152:
1150:
1149:
1144:
1129:
1127:
1126:
1121:
1098:
1096:
1095:
1090:
1088:
1083:
1072:
1063:
1061:
1060:
1055:
1037:
1035:
1034:
1029:
1024:
1016:
996:
994:
993:
988:
970:
968:
967:
962:
944:
942:
941:
936:
912:
910:
909:
904:
902:
897:
886:
877:
875:
874:
869:
851:
849:
848:
843:
825:
823:
822:
817:
812:
804:
784:
782:
781:
776:
758:
756:
755:
750:
732:
730:
729:
724:
698:
696:
695:
690:
672:
670:
669:
664:
646:
644:
643:
638:
612:
610:
609:
604:
583:
581:
580:
575:
557:
555:
554:
549:
534:
532:
531:
526:
508:
506:
505:
500:
476:
475:
465:
464:
455:
448:
441:
385:
377:
375:
374:
369:
343:
341:
340:
335:
312:
304:
299:
291:
271:
269:
268:
263:
261:
253:
236:
228:
206:
192:
190:
189:
184:
158:
146:
142:
136:
129:
125:
122:
116:
114:
73:
49:
41:
21:
8797:
8796:
8792:
8791:
8790:
8788:
8787:
8786:
8762:
8761:
8760:
8755:
8747:
8745:
8739:
8722:
8719:
8718:
8702:
8685:
8682:
8681:
8656:
8653:
8652:
8631:
8628:
8627:
8602:
8599:
8598:
8582:
8555:
8552:
8551:
8526:
8523:
8522:
8497:
8494:
8493:
8477:
8460:
8457:
8456:
8431:
8428:
8427:
8402:
8399:
8398:
8377:
8374:
8373:
8357:
8348:
8331:
8328:
8327:
8311:
8305:
8263:
8258:
8248:
8246:
8240:
8236:
8226:
8224:
8221:
8215:
8211:
8190:
8186:
8162:
8158:
8136:
8132:
8125:
8117:. p. 496.
8105:
8101:
8090:
8086:
8071:
8067:
8057:
8055:
8046:
8042:
8019:
8015:
7984:(12): 713–723.
7973:
7964:
7960:
7939:
7935:
7914:
7910:
7898:
7894:
7879:
7875:
7864:
7860:
7849:
7845:
7832:
7825:
7814:
7812:
7808:
7799:
7797:
7783:
7772:
7765:
7747:
7743:
7736:
7705:
7697:
7690:
7680:
7678:
7662:
7658:
7654:
7649:
7645:XOR linked list
7566:
7553:
7549:
7546:
7541:
7540:
7536:
7532:
7529:
7524:
7523:
7507:
7504:
7503:
7495:
7492:
7442:
7440:
7435:
7433:
7428:
7426:
7421:
7419:
7414:
7412:
7407:
7405:
7400:XOR is used in
7373:Feistel network
7342:parity function
7293:
7290:
7289:
7255:
7251:
7246:
7238:
7233:
7228:
7225:
7224:
7204:
7202:
7197:
7195:
7190:
7188:
7152:
7148:
7143:
7135:
7130:
7125:
7122:
7121:
7099:
7093:
7077:
7026:
7023:
7022:
6991:
6988:
6987:
6955:
6952:
6951:
6933:
6930:
6929:
6905:
6902:
6901:
6878:
6875:
6874:
6845:
6842:
6841:
6829:
6821:
6812:
6778:
6775:
6774:
6741:
6738:
6737:
6719:
6716:
6715:
6691:
6688:
6687:
6679:
6648:
6645:
6644:
6611:
6608:
6607:
6589:
6586:
6585:
6561:
6558:
6557:
6549:
6519:
6516:
6515:
6490:
6487:
6486:
6460:
6457:
6456:
6417:
6414:
6413:
6396:
6393:
6392:
6363:
6360:
6359:
6337:
6334:
6333:
6309:
6306:
6305:
6298:
6264:
6261:
6260:
6234:
6231:
6230:
6199:
6196:
6195:
6163:
6160:
6159:
6141:
6138:
6137:
6113:
6110:
6109:
6091:
6088:
6087:
6063:
6060:
6059:
6036:
6033:
6032:
6009:
6006:
6005:
5998:
5914:
5911:
5910:
5902:
5851:
5848:
5847:
5822:
5819:
5818:
5792:
5789:
5788:
5745:
5742:
5741:
5712:
5709:
5708:
5673:
5670:
5669:
5640:
5637:
5636:
5614:
5611:
5610:
5580:
5577:
5576:
5541:
5538:
5537:
5517:
5514:
5513:
5506:
5472:
5469:
5468:
5435:
5432:
5431:
5413:
5410:
5409:
5385:
5382:
5381:
5374:
5366:
5338:
5333:
5330:
5329:
5307:
5302:
5299:
5298:
5276:
5273:
5272:
5256:
5253:
5252:
5236:
5233:
5232:
5173:
5155:
5152:
5151:
5135:
5132:
5131:
5124:Polish notation
5107:
5104:
5103:
5100:Jan Łukasiewicz
5073:
5070:
5069:
5066:prefix operator
5049:
5046:
5045:
5019:
5016:
5015:
4997:
4994:
4993:
4964:
4961:
4960:
4939:
4936:
4935:
4917:
4914:
4913:
4893:
4890:
4889:
4869:
4866:
4865:
4817:
4814:
4813:
4793:
4790:
4789:
4771:
4768:
4767:
4735:
4732:
4731:
4707:
4704:
4703:
4680:
4678:
4675:
4674:
4653:
4651:
4648:
4647:
4631:
4628:
4627:
4611:
4608:
4607:
4591:
4588:
4587:
4571:
4568:
4567:
4540:
4535:
4532:
4531:
4506:
4504:
4501:
4500:
4482:
4479:
4478:
4462:
4459:
4458:
4436:
4433:
4432:
4410:
4407:
4406:
4390:
4387:
4386:
4366:
4363:
4362:
4355:
4286:
4260:
4255:
4254:
4252:
4249:
4248:
4223:
4222:
4206:
4189:
4184:
4166:
4165:
4149:
4132:
4127:
4108:
4106:
4103:
4102:
4082:
4077:
4076:
4074:
4071:
4070:
4053:
4048:
4047:
4045:
4042:
4041:
4025:
4022:
4021:
4005:
4002:
4001:
3970:
3967:
3966:
3948:
3943:
3942:
3940:
3937:
3936:
3906:
3903:
3902:
3886:
3883:
3882:
3866:
3863:
3862:
3815:
3812:
3811:
3756:
3753:
3752:
3712:
3709:
3708:
3685:
3682:
3681:
3661:
3658:
3657:
3651:
3583:
3580:
3579:
3572:
3553:
3552:
3536:
3534:
3514:
3509:
3489:
3484:
3467:
3460:
3459:
3446:
3433:
3413:
3408:
3385:
3380:
3363:
3356:
3355:
3342:
3329:
3324:
3319:
3299:
3294:
3274:
3269:
3256:
3254:
3251:
3250:
3204:
3203:
3162:
3157:
3144:
3142:
3139:
3138:
3118:
3117:
3067:
3062:
3049:
3047:
3044:
3043:
3018:
3015:
3014:
2997:
2996:
2979:
2974:
2954:
2947:
2946:
2929:
2924:
2901:
2894:
2893:
2849:
2844:
2794:
2787:
2786:
2754:
2749:
2714:
2707:
2706:
2686:
2681:
2661:
2656:
2643:
2641:
2638:
2637:
2618:
2615:
2614:
2598:
2595:
2594:
2578:
2575:
2574:
2557:
2556:
2515:
2510:
2497:
2495:
2492:
2491:
2466:
2463:
2462:
2445:
2444:
2406:
2401:
2388:
2386:
2383:
2382:
2363:
2360:
2359:
2339:
2336:
2335:
2334:("logical or",
2315:
2312:
2311:
2285:
2282:
2281:
2259:
2254:
2251:
2250:
2228:
2225:
2224:
2217:
2144:
2141:
2140:
2123:
2120:
2119:
2102:
2099:
2098:
2071:
2068:
2067:
2041:
2020:
2017:
2016:
2000:
1997:
1996:
1980:
1977:
1976:
1971:
1949:
1947:
1944:
1943:
1922:
1920:
1917:
1916:
1894:
1893:
1891:
1888:
1887:
1860:
1856:
1834:
1830:
1817:
1803:
1799:
1783:
1769:
1765:
1759:infix operators
1742:
1739:
1738:
1658:
1655:
1654:
1640:
1599:
1566:
1537:Boolean algebra
1532:Predicate logic
1489:
1486:
1485:
1463:
1460:
1459:
1437:
1434:
1433:
1403:
1400:
1399:
1375:
1370:
1367:
1366:
1336:
1328:
1325:
1324:
1294:
1291:
1290:
1268:
1265:
1264:
1242:
1239:
1238:
1216:
1213:
1212:
1185:
1182:
1181:
1160:
1158:
1155:
1154:
1135:
1132:
1131:
1112:
1109:
1108:
1073:
1071:
1069:
1066:
1065:
1043:
1040:
1039:
1015:
1010:
1007:
1006:
976:
973:
972:
950:
947:
946:
924:
921:
920:
887:
885:
883:
880:
879:
857:
854:
853:
831:
828:
827:
803:
798:
795:
794:
764:
761:
760:
738:
735:
734:
712:
709:
708:
678:
675:
674:
652:
649:
648:
626:
623:
622:
589:
586:
585:
563:
560:
559:
540:
537:
536:
514:
511:
510:
488:
485:
484:
459:
384:Post's lattices
383:
357:
354:
353:
303:
290:
285:
282:
281:
252:
227:
225:
222:
221:
172:
169:
168:
137:
126:
120:
117:
74:
72:
62:
50:
39:
28:
23:
22:
15:
12:
11:
5:
8795:
8785:
8784:
8779:
8774:
8757:
8756:
8744:
8741:
8740:
8738:
8737:
8726:
8707:
8704:
8703:
8701:
8700:
8689:
8671:
8660:
8646:
8635:
8620:Nonimplication
8617:
8606:
8587:
8584:
8583:
8581:
8580:
8577:Digital buffer
8570:
8559:
8541:
8530:
8512:
8501:
8482:
8479:
8478:
8476:
8475:
8464:
8446:
8435:
8417:
8406:
8392:
8381:
8362:
8359:
8358:
8351:
8349:
8347:
8346:
8335:
8316:
8313:
8312:
8304:
8303:
8296:
8289:
8281:
8275:
8274:
8269:
8262:
8261:External links
8259:
8257:
8256:
8234:
8209:
8184:
8156:
8130:
8123:
8099:
8084:
8065:
8040:
8029:(3): 288–309.
8013:
7967:Shannon, C. E.
7958:
7949:Translated as
7933:
7908:
7892:
7873:
7858:
7843:
7823:
7806:
7770:
7763:
7741:
7734:
7710:Translated as
7688:
7655:
7653:
7650:
7648:
7647:
7642:
7637:
7632:
7627:
7622:
7617:
7612:
7607:
7602:
7597:
7592:
7587:
7582:
7577:
7567:
7565:
7562:
7511:
7491:
7488:
7484:alpha channels
7476:bounding boxes
7346:
7345:
7340:returned by a
7334:if and only if
7321:
7318:
7315:
7312:
7309:
7306:
7303:
7300:
7297:
7282:
7279:
7258:
7254:
7249:
7245:
7241:
7236:
7232:
7214:
7213:
7186:
7183:
7180:
7177:
7155:
7151:
7146:
7142:
7138:
7133:
7129:
7095:Main article:
7092:
7089:
7076:
7073:
7054:
7053:
7050:
7049:
7042:
7030:
7019:
7012:
6998:
6985:
6977:
6976:
6962:
6949:
6937:
6926:
6912:
6899:
6885:
6872:
6858:
6855:
6852:
6830:
6827:
6825:
6822:
6815:
6813:
6809:Walsh spectrum
6807:
6805:
6802:
6801:
6794:
6782:
6771:
6763:
6762:
6751:
6748:
6745:
6735:
6723:
6712:
6701:
6698:
6695:
6680:
6677:
6675:
6672:
6671:
6664:
6652:
6641:
6633:
6632:
6621:
6618:
6615:
6605:
6593:
6582:
6571:
6568:
6565:
6550:
6547:
6545:
6542:
6541:
6534:
6523:
6513:
6506:
6494:
6483:
6476:
6464:
6453:
6445:
6444:
6433:
6430:
6427:
6424:
6421:
6411:
6400:
6390:
6379:
6376:
6373:
6370:
6367:
6357:
6355:
6353:
6341:
6330:
6319:
6316:
6313:
6299:
6293:
6291:
6288:
6287:
6280:
6268:
6257:
6250:
6238:
6227:
6220:
6206:
6193:
6185:
6184:
6170:
6157:
6145:
6134:
6120:
6107:
6095:
6084:
6070:
6057:
6043:
6030:
6016:
5999:
5993:
5991:
5972:
5969:
5966:
5963:
5960:
5957:
5954:
5951:
5948:
5945:
5942:
5939:
5936:
5933:
5930:
5927:
5924:
5921:
5918:
5903:
5899:Distributivity
5897:
5895:
5892:
5891:
5884:
5864:
5845:
5838:
5826:
5815:
5808:
5796:
5785:
5778:
5758:
5739:
5731:
5730:
5719:
5706:
5686:
5667:
5656:
5653:
5650:
5647:
5644:
5634:
5632:
5630:
5618:
5607:
5596:
5593:
5590:
5587:
5584:
5574:
5554:
5535:
5524:
5507:
5501:
5499:
5496:
5495:
5488:
5476:
5465:
5457:
5456:
5445:
5442:
5439:
5429:
5417:
5406:
5395:
5392:
5389:
5375:
5369:
5365:
5362:
5361:
5360:
5348:
5345:
5341:
5337:
5317:
5314:
5310:
5306:
5286:
5283:
5280:
5260:
5240:
5225:
5184:to denote the
5171:
5159:
5139:
5111:
5094:) was used by
5083:
5080:
5077:
5053:
5043:
5038:, was used by
5023:
5013:
5001:
4981:Claude Shannon
4968:
4958:
4946:
4943:
4933:
4921:
4897:
4873:
4853:
4850:
4847:
4843:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4810:Giuseppe Peano
4797:
4787:
4775:
4764:Giuseppe Peano
4739:
4728:Ernst Schröder
4726:, was used by
4711:
4701:
4687:
4684:
4660:
4657:
4646:", i.e., used
4635:
4615:
4595:
4575:
4555:
4552:
4547:
4544:
4539:
4513:
4510:
4498:
4486:
4466:
4446:
4443:
4440:
4420:
4417:
4414:
4394:
4370:
4354:
4351:
4317:
4316:
4313:
4298:
4297:
4285:
4282:
4263:
4258:
4237:
4236:
4220:
4217:
4213:
4210:
4205:
4202:
4199:
4196:
4193:
4190:
4188:
4185:
4183:
4180:
4177:
4174:
4171:
4168:
4167:
4163:
4160:
4156:
4153:
4148:
4145:
4142:
4139:
4136:
4133:
4131:
4128:
4126:
4123:
4120:
4117:
4114:
4111:
4110:
4085:
4080:
4056:
4051:
4029:
4009:
3986:
3983:
3980:
3977:
3974:
3951:
3946:
3922:
3919:
3916:
3913:
3910:
3901:over elements
3890:
3870:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3689:
3665:
3650:
3647:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3571:
3568:
3567:
3566:
3551:
3546:
3542:
3539:
3533:
3530:
3527:
3524:
3521:
3518:
3515:
3513:
3510:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3488:
3485:
3483:
3480:
3477:
3474:
3471:
3468:
3466:
3463:
3461:
3458:
3453:
3450:
3445:
3440:
3437:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3412:
3409:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3384:
3381:
3379:
3376:
3373:
3370:
3367:
3364:
3362:
3359:
3357:
3354:
3349:
3346:
3341:
3336:
3333:
3328:
3325:
3323:
3320:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3298:
3295:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3273:
3270:
3268:
3265:
3262:
3259:
3258:
3218:
3217:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3161:
3158:
3156:
3153:
3150:
3147:
3146:
3132:
3131:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3066:
3063:
3061:
3058:
3055:
3052:
3051:
3028:
3025:
3022:
3011:
3010:
2995:
2992:
2989:
2986:
2983:
2980:
2978:
2975:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2953:
2950:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2928:
2925:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2900:
2897:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2848:
2845:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2793:
2790:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2753:
2750:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2713:
2710:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2685:
2682:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2660:
2657:
2655:
2652:
2649:
2646:
2645:
2622:
2602:
2582:
2571:
2570:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2514:
2511:
2509:
2506:
2503:
2500:
2499:
2476:
2473:
2470:
2459:
2458:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2405:
2402:
2400:
2397:
2394:
2391:
2390:
2378:) as follows:
2367:
2343:
2319:
2295:
2292:
2289:
2269:
2266:
2262:
2258:
2238:
2235:
2232:
2221:if and only if
2216:
2213:
2210:
2209:
2206:
2203:
2199:
2198:
2195:
2192:
2188:
2187:
2184:
2181:
2177:
2176:
2173:
2170:
2166:
2165:
2154:
2151:
2148:
2138:
2127:
2117:
2106:
2081:
2078:
2075:
2040:
2037:
2024:
2004:
1984:
1956:
1953:
1929:
1926:
1901:
1898:
1746:
1726:are true. XOR
1674:
1671:
1668:
1665:
1662:
1642:
1641:
1639:
1638:
1631:
1624:
1616:
1613:
1612:
1601:
1600:
1598:
1597:
1592:
1587:
1582:
1576:
1573:
1572:
1568:
1567:
1565:
1564:
1559:
1554:
1549:
1547:Truth function
1544:
1539:
1534:
1529:
1523:
1520:
1519:
1515:
1514:
1511:
1510:
1499:
1496:
1493:
1473:
1470:
1467:
1447:
1444:
1441:
1431:
1425:
1424:
1413:
1410:
1407:
1387:
1382:
1379:
1374:
1364:
1358:
1357:
1346:
1332:
1322:
1316:
1315:
1304:
1301:
1298:
1278:
1275:
1272:
1252:
1249:
1246:
1226:
1223:
1220:
1210:
1204:
1203:
1192:
1189:
1167:
1164:
1142:
1139:
1119:
1116:
1106:
1100:
1099:
1086:
1082:
1079:
1076:
1053:
1050:
1047:
1027:
1022:
1019:
1014:
1004:
998:
997:
986:
983:
980:
960:
957:
954:
934:
931:
928:
918:
914:
913:
900:
896:
893:
890:
867:
864:
861:
841:
838:
835:
815:
810:
807:
802:
792:
786:
785:
774:
771:
768:
748:
745:
742:
722:
719:
716:
706:
700:
699:
688:
685:
682:
662:
659:
656:
636:
633:
630:
620:
614:
613:
602:
599:
596:
593:
573:
570:
567:
547:
544:
524:
521:
518:
498:
495:
492:
482:
472:
471:
461:
460:
458:
457:
450:
443:
435:
432:
431:
428:
424:
423:
420:
414:
413:
410:
404:
403:
400:
396:
395:
392:
388:
387:
379:
378:
367:
364:
361:
351:
345:
344:
333:
330:
327:
324:
321:
318:
315:
310:
307:
302:
297:
294:
289:
279:
273:
272:
259:
256:
251:
248:
245:
242:
239:
234:
231:
219:
213:
212:
208:
207:
200:
194:
193:
182:
179:
176:
166:
160:
159:
151:
150:
139:
138:
80:"Exclusive or"
53:
51:
44:
26:
9:
6:
4:
3:
2:
8794:
8783:
8780:
8778:
8775:
8773:
8770:
8769:
8767:
8754:
8742:
8716:
8712:
8711:Contradiction
8709:
8708:
8705:
8687:
8679:
8675:
8672:
8658:
8650:
8647:
8633:
8625:
8621:
8618:
8596:
8592:
8589:
8588:
8585:
8578:
8574:
8571:
8549:
8545:
8544:Biconditional
8542:
8528:
8520:
8516:
8513:
8491:
8487:
8484:
8483:
8480:
8462:
8454:
8450:
8447:
8425:
8421:
8418:
8396:
8393:
8371:
8367:
8364:
8363:
8360:
8355:
8325:
8321:
8318:
8317:
8314:
8310:
8302:
8297:
8295:
8290:
8288:
8283:
8282:
8279:
8273:
8270:
8268:
8267:All About XOR
8265:
8264:
8245:
8238:
8220:
8213:
8204:
8203:
8198:
8195:
8188:
8180:
8179:
8174:
8170:
8166:
8160:
8152:
8148:
8144:
8140:
8134:
8126:
8120:
8116:
8112:
8111:
8103:
8095:
8088:
8081:(1): 274–304.
8080:
8076:
8069:
8053:
8052:
8044:
8036:
8032:
8028:
8024:
8017:
8009:
8005:
8000:
7995:
7991:
7987:
7983:
7979:
7972:
7968:
7962:
7954:
7946:
7945:
7937:
7929:
7928:
7922:Reprinted in
7919:
7912:
7903:
7896:
7888:
7884:
7877:
7869:
7862:
7854:
7847:
7839:
7838:
7830:
7828:
7819:
7810:
7795:
7791:
7790:"Disjunction"
7787:
7781:
7779:
7777:
7775:
7766:
7764:9781420070033
7760:
7756:
7752:
7745:
7737:
7731:
7727:
7723:
7718:
7717:
7704:
7703:
7695:
7693:
7677:
7673:
7672:
7667:
7660:
7656:
7646:
7643:
7641:
7638:
7636:
7633:
7631:
7628:
7626:
7623:
7621:
7620:Logical value
7618:
7616:
7615:Logical graph
7613:
7611:
7608:
7606:
7603:
7601:
7598:
7596:
7593:
7591:
7588:
7586:
7583:
7581:
7578:
7576:
7572:
7569:
7568:
7561:
7559:
7509:
7501:
7487:
7485:
7481:
7477:
7473:
7468:
7466:
7462:
7458:
7456:
7451:
7447:
7403:
7398:
7396:
7392:
7391:entropy pools
7387:
7385:
7381:
7376:
7374:
7370:
7366:
7361:
7357:
7355:
7351:
7343:
7339:
7335:
7319:
7316:
7313:
7310:
7307:
7304:
7301:
7298:
7295:
7287:
7283:
7280:
7277:
7276:
7275:
7272:
7256:
7243:
7239:
7223:
7219:
7211:
7187:
7184:
7181:
7178:
7175:
7174:
7173:
7153:
7140:
7136:
7119:
7115:
7111:
7107:
7103:
7098:
7086:
7081:
7072:
7070:
7067:
7063:
7059:
7047:
7043:
7020:
7017:
7013:
6996:
6986:
6983:
6979:
6978:
6960:
6950:
6927:
6910:
6900:
6883:
6873:
6856:
6853:
6850:
6840:
6839:
6835:
6831:
6823:
6819:
6810:
6799:
6795:
6772:
6769:
6765:
6764:
6749:
6746:
6743:
6736:
6713:
6699:
6696:
6693:
6686:
6685:
6681:
6669:
6665:
6650:
6642:
6639:
6635:
6634:
6619:
6616:
6613:
6606:
6591:
6583:
6569:
6566:
6563:
6556:
6555:
6551:
6539:
6535:
6514:
6511:
6507:
6484:
6481:
6477:
6462:
6454:
6451:
6447:
6446:
6428:
6425:
6422:
6412:
6391:
6374:
6371:
6368:
6358:
6356:
6354:
6339:
6331:
6317:
6311:
6304:
6303:
6300:
6296:
6285:
6281:
6266:
6258:
6255:
6251:
6228:
6225:
6221:
6204:
6194:
6191:
6187:
6186:
6168:
6158:
6143:
6135:
6118:
6108:
6085:
6068:
6058:
6041:
6031:
6014:
6004:
6003:
6000:
5996:
5989:
5986:
5967:
5964:
5961:
5955:
5949:
5946:
5943:
5937:
5931:
5928:
5925:
5919:
5916:
5908:
5904:
5900:
5889:
5885:
5862:
5846:
5843:
5839:
5816:
5813:
5809:
5786:
5783:
5779:
5756:
5740:
5737:
5733:
5732:
5717:
5707:
5684:
5668:
5651:
5648:
5645:
5635:
5633:
5631:
5608:
5591:
5588:
5585:
5575:
5552:
5536:
5522:
5512:
5511:
5508:
5504:
5503:Associativity
5493:
5489:
5466:
5463:
5459:
5458:
5443:
5440:
5437:
5430:
5407:
5393:
5390:
5387:
5380:
5379:
5376:
5372:
5371:Commutativity
5367:
5346:
5343:
5339:
5335:
5315:
5312:
5308:
5304:
5284:
5281:
5278:
5258:
5238:
5230:
5226:
5223:
5219:
5215:
5211:
5207:
5203:
5199:
5195:
5191:
5187:
5183:
5179:
5172:
5157:
5137:
5129:
5125:
5109:
5101:
5097:
5081:
5078:
5075:
5067:
5051:
5044:
5041:
5040:Alonzo Church
5037:
5021:
5014:
4999:
4990:
4986:
4982:
4966:
4959:
4944:
4941:
4934:
4919:
4911:
4895:
4887:
4871:
4851:
4848:
4845:
4841:
4837:
4834:
4831:
4828:
4825:
4822:
4819:
4811:
4795:
4788:
4773:
4765:
4761:
4757:
4753:
4737:
4729:
4725:
4709:
4702:
4682:
4655:
4633:
4613:
4593:
4573:
4553:
4550:
4542:
4537:
4529:
4508:
4499:
4484:
4464:
4444:
4441:
4438:
4418:
4415:
4412:
4392:
4384:
4368:
4361:
4360:
4359:
4350:
4348:
4343:
4341:
4337:
4333:
4329:
4325:
4322:
4314:
4311:
4310:
4309:
4307:
4303:
4295:
4294:
4293:
4291:
4281:
4279:
4261:
4246:
4242:
4215:
4211:
4203:
4200:
4197:
4194:
4191:
4181:
4178:
4175:
4172:
4169:
4158:
4154:
4146:
4143:
4140:
4137:
4134:
4124:
4121:
4118:
4115:
4112:
4101:
4100:
4099:
4083:
4054:
4027:
4007:
3998:
3981:
3978:
3975:
3964:
3949:
3917:
3914:
3911:
3888:
3868:
3860:
3859:abelian group
3856:
3838:
3835:
3829:
3826:
3823:
3808:
3806:
3802:
3798:
3779:
3776:
3770:
3767:
3764:
3735:
3732:
3726:
3723:
3720:
3705:
3703:
3687:
3679:
3663:
3656:
3653:Although the
3646:
3633:
3630:
3624:
3621:
3615:
3612:
3609:
3597:
3594:
3591:
3577:
3540:
3537:
3525:
3522:
3519:
3511:
3503:
3500:
3497:
3486:
3478:
3475:
3472:
3464:
3448:
3443:
3435:
3424:
3421:
3418:
3410:
3402:
3396:
3393:
3382:
3374:
3371:
3368:
3360:
3352:
3344:
3339:
3331:
3326:
3321:
3313:
3310:
3307:
3296:
3288:
3282:
3279:
3271:
3266:
3263:
3260:
3249:
3248:
3247:
3244:
3242:
3238:
3234:
3230:
3225:
3223:
3197:
3191:
3188:
3179:
3173:
3170:
3167:
3159:
3154:
3151:
3148:
3137:
3136:
3135:
3108:
3102:
3099:
3090:
3084:
3081:
3078:
3064:
3059:
3056:
3053:
3042:
3041:
3040:
3026:
3023:
3020:
2990:
2987:
2984:
2976:
2968:
2965:
2962:
2951:
2940:
2937:
2934:
2926:
2918:
2912:
2909:
2898:
2884:
2881:
2878:
2869:
2863:
2860:
2857:
2846:
2835:
2829:
2826:
2817:
2811:
2805:
2802:
2791:
2780:
2777:
2771:
2765:
2762:
2751:
2743:
2737:
2731:
2725:
2722:
2711:
2700:
2697:
2694:
2683:
2675:
2669:
2666:
2658:
2653:
2650:
2647:
2636:
2635:
2634:
2620:
2600:
2550:
2547:
2544:
2535:
2529:
2523:
2520:
2512:
2507:
2504:
2501:
2490:
2489:
2488:
2474:
2471:
2468:
2438:
2435:
2432:
2423:
2417:
2414:
2411:
2403:
2398:
2395:
2392:
2381:
2380:
2379:
2357:
2341:
2333:
2317:
2309:
2293:
2290:
2287:
2267:
2264:
2260:
2256:
2236:
2233:
2230:
2222:
2207:
2204:
2201:
2200:
2196:
2193:
2190:
2189:
2185:
2182:
2179:
2178:
2174:
2171:
2168:
2167:
2152:
2149:
2146:
2139:
2125:
2118:
2104:
2097:
2096:
2093:
2079:
2076:
2073:
2065:
2054:
2050:
2045:
2036:
2022:
2002:
1982:
1974:
1954:
1951:
1924:
1899:
1896:
1885:
1881:
1875:
1852:
1826:
1795:
1763:
1760:
1744:
1736:
1731:
1729:
1725:
1720:
1718:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1672:
1669:
1666:
1663:
1660:
1652:
1648:
1637:
1632:
1630:
1625:
1623:
1618:
1617:
1615:
1614:
1611:
1603:
1602:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1580:Digital logic
1578:
1577:
1575:
1574:
1570:
1569:
1563:
1562:Scope (logic)
1560:
1558:
1555:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1535:
1533:
1530:
1528:
1525:
1524:
1522:
1521:
1517:
1516:
1497:
1491:
1471:
1468:
1465:
1445:
1439:
1432:
1430:
1427:
1426:
1411:
1408:
1405:
1385:
1380:
1377:
1372:
1365:
1363:
1360:
1359:
1344:
1330:
1323:
1321:
1318:
1317:
1302:
1299:
1296:
1276:
1273:
1270:
1250:
1247:
1244:
1224:
1221:
1218:
1211:
1209:
1206:
1205:
1190:
1187:
1162:
1140:
1137:
1117:
1107:
1105:
1102:
1101:
1080:
1077:
1074:
1051:
1045:
1025:
1017:
1012:
1005:
1003:
1000:
999:
984:
981:
978:
958:
955:
952:
932:
929:
926:
919:
917:nonequivalent
916:
915:
894:
891:
888:
865:
862:
859:
839:
833:
813:
805:
800:
793:
791:
788:
787:
772:
766:
746:
743:
740:
720:
714:
707:
705:
702:
701:
686:
680:
660:
654:
634:
631:
628:
621:
619:
616:
615:
600:
591:
571:
565:
545:
542:
522:
519:
516:
496:
493:
490:
483:
481:
478:
477:
474:
473:
470:
467:
466:
456:
451:
449:
444:
442:
437:
436:
433:
429:
425:
421:
419:
415:
411:
409:
405:
401:
397:
393:
389:
386:
380:
365:
362:
359:
352:
350:
346:
328:
325:
322:
316:
305:
300:
292:
280:
278:
274:
254:
249:
246:
243:
240:
237:
229:
220:
218:
214:
209:
205:
201:
199:
195:
177:
167:
165:
161:
157:
152:
147:
135:
132:
124:
113:
110:
106:
103:
99:
96:
92:
89:
85:
82: –
81:
77:
76:Find sources:
70:
66:
60:
59:
54:This article
52:
48:
43:
42:
37:
33:
19:
8591:Joint denial
8515:Exclusive or
8514:
8247:. Retrieved
8237:
8225:. Retrieved
8212:
8200:
8187:
8177:
8159:
8146:
8142:
8133:
8109:
8102:
8093:
8087:
8078:
8074:
8068:
8056:. Retrieved
8050:
8043:
8026:
8022:
8016:
7999:1721.1/11173
7981:
7977:
7961:
7952:
7943:
7936:
7926:
7917:
7911:
7901:
7895:
7886:
7876:
7867:
7861:
7852:
7846:
7836:
7817:
7809:
7798:. Retrieved
7793:
7786:Aloni, Maria
7754:
7744:
7715:
7701:
7679:. Retrieved
7669:
7659:
7600:Inclusive or
7550:CIRCLED PLUS
7537:⊻
7493:
7469:
7459:
7452:
7448:
7399:
7388:
7377:
7369:one-time pad
7365:cryptography
7362:
7358:
7347:
7273:
7222:vector space
7217:
7215:
7171:
7110:exclusive or
7109:
7062:exclusive or
7061:
7055:
6811:: (2,0,0,−2)
6295:Monotonicity
5231:of two sets
4979:was used by
4909:
4885:
4808:was used by
4760:Hugh MacColl
4752:George Boole
4566:to express "
4526:was used by
4383:George Boole
4381:was used by
4356:
4347:soit... soit
4346:
4344:
4318:
4302:felicitously
4299:
4287:
4238:
3999:
3854:
3809:
3707:The systems
3706:
3652:
3573:
3245:
3226:
3219:
3133:
3012:
2572:
2460:
2218:
2061:
2049:Walsh matrix
1970:
1883:
1879:
1761:
1732:
1727:
1721:
1700:
1696:
1692:
1689:Exclusive or
1688:
1687:
1651:Venn diagram
1571:Applications
1361:
399:1-preserving
391:0-preserving
211:Normal forms
144:Exclusive or
127:
118:
108:
101:
94:
87:
75:
63:Please help
58:verification
55:
18:Exclusive-or
8772:Dichotomies
8674:Conjunction
8624:NIMPLY gate
8449:Disjunction
8420:Implication
7185:0 XOR 0 = 0
7182:0 XOR 1 = 1
7179:1 XOR 0 = 1
7176:1 XOR 1 = 0
6828:Involution:
5995:Idempotency
5128:connectives
5036:equivalence
4888:; the sign
4864:. The sign
4724:equivalence
4332:cancellable
4020:with 0 and
3702:disjunction
3678:conjunction
2354:), and the
2332:disjunction
2064:truth table
1757:and by the
1542:Truth table
277:Conjunctive
217:Disjunctive
164:Truth table
8766:Categories
8424:IMPLY gate
7800:2020-09-03
7635:XOR cipher
7605:Involution
7386:function.
7338:parity bit
7085:logic gate
6834:involution
5364:Properties
4812:in 1894: "
4340:entailment
4245:polynomial
3578:, we get:
3237:antecedent
2039:Definition
1735:symbolized
618:equivalent
198:Logic gate
91:newspapers
8782:Semantics
8725:⊥
8688:∧
8659:↚
8634:↛
8605:↓
8573:Statement
8558:↔
8548:XNOR gate
8500:¬
8463:∨
8434:→
8405:←
8380:↑
8370:NAND gate
8334:⊤
8320:Tautology
8227:28 August
8202:MathWorld
7671:MathWorld
7575:(Paradox)
7510:↮
7490:Encodings
7432:is lost,
7317:⊕
7311:⊕
7305:⊕
7299:⊕
7056:If using
7029:⇔
6997:⊕
6936:⇔
6884:⊕
6854:⊕
6818:linearity
6781:⇒
6747:∨
6722:⇒
6697:⊕
6651:⇏
6617:⊕
6592:⇏
6567:∧
6522:→
6493:⇔
6463:⇏
6426:⊕
6399:→
6372:⊕
6340:⇏
6315:→
6267:⇎
6237:⇔
6205:⊕
6144:⇎
6094:⇔
6042:⊕
5965:∧
5956:⊕
5947:∧
5929:⊕
5920:∧
5863:⊕
5825:⇔
5795:⇔
5757:⊕
5685:⊕
5649:⊕
5617:⇔
5589:⊕
5553:⊕
5475:⇔
5441:⊕
5416:⇔
5391:⊕
5344:
5340:△
5313:
5309:▽
5282:⊖
5082:ψ
5079:ϕ
4967:⊕
4945:∨
4942:∨
4920:∪
4896:∪
4872:∘
4849:−
4842:∪
4835:−
4823:∘
4796:∘
4686:¯
4683:∨
4659:¯
4656:∨
4606:" or "No
4551:
4546:¯
4543:∨
4512:¯
4509:∨
4485:∨
4328:semantics
4321:pragmatic
4187:⇔
4179:⊕
4144:⋅
4130:⇔
4122:∧
3982:∨
3976:∧
3889:⊕
3869:∧
3839:⊕
3780:∨
3736:∧
3688:∨
3664:∧
3655:operators
3628:¬
3625:↮
3619:⇔
3613:↮
3607:¬
3604:⇔
3595:↮
3586:¬
3545:¯
3501:∧
3492:¬
3487:∧
3476:∨
3452:¯
3439:¯
3400:¬
3397:∨
3391:¬
3383:∧
3372:∨
3348:¯
3335:¯
3311:∧
3305:¬
3297:∨
3286:¬
3283:∧
3264:↮
3195:¬
3192:∨
3186:¬
3180:∧
3171:∨
3152:↮
3106:¬
3103:∧
3097:¬
3091:∨
3082:∧
3070:¬
3057:↮
3024:↮
2988:∨
2977:∧
2966:∧
2957:¬
2938:∨
2927:∧
2916:¬
2913:∨
2907:¬
2882:∨
2876:¬
2870:∧
2861:∨
2847:∧
2833:¬
2830:∨
2824:¬
2818:∧
2809:¬
2806:∨
2778:∨
2769:¬
2766:∧
2752:∧
2741:¬
2738:∨
2729:¬
2726:∧
2698:∧
2692:¬
2684:∨
2673:¬
2670:∧
2651:↮
2621:∨
2601:∧
2581:¬
2548:∧
2542:¬
2536:∨
2527:¬
2524:∧
2505:↮
2472:↮
2436:∧
2427:¬
2424:∧
2415:∨
2396:↮
2366:¬
2342:∨
2318:∧
2265:
2234:↮
2150:⊕
2077:⊕
2003:↮
1983:⊕
1955:_
1952:∨
1928:¯
1925:∨
1900:˙
1897:∨
1670:⊕
1664:⊕
1495:←
1469:⊂
1443:⇐
1409:⊕
1381:_
1378:∨
1300:∥
1274:∣
1222:∨
1188:∼
1166:¯
1138:−
1115:¬
1085:¯
1049:↓
1021:¯
1018:∨
982:↮
899:¯
892:⋅
863:∣
837:↑
809:¯
806:∧
770:→
744:⊃
718:⇒
684:⇋
658:⇔
632:≡
598:&
595:&
569:&
520:⋅
494:∧
427:Self-dual
363:⊕
317:⋅
309:¯
296:¯
258:¯
250:⋅
238:⋅
233:¯
8678:AND gate
8595:NOR gate
8529:↮
8519:XOR gate
8490:NOT gate
8486:Negation
8249:23 March
8171:(1978).
8141:(1929).
8008:51638483
7969:(1938).
7788:(2016).
7640:XOR gate
7564:See also
7547:⊕
7530:⊻
7441:11110000
7434:10011100
7427:01101100
7420:11110000
7413:01101100
7406:10011100
7354:XOR gate
7332:is true
7066:addition
5042:in 1944.
5022:≢
4774:≠
4710:≠
2356:negation
2053:variadic
2023:≢
1728:excludes
1724:operands
1610:Category
1429:converse
956:⇎
930:≢
408:Monotone
121:May 2013
32:XOR gate
8680:)
8676: (
8626:)
8622: (
8597:)
8593: (
8575: (
8550:)
8546: (
8521:)
8517: (
8492:)
8488: (
8455:)
8453:OR gate
8451: (
8426:)
8422: (
8372:)
8368: (
8307:Common
7681:17 June
7630:Rule 90
7585:Ampheck
7480:cursors
5186:bitwise
4586:is-not
3797:monoids
2330:), the
704:implies
105:scholar
8717:
8651:
8397:
8326:
8121:
8058:7 July
8006:
7761:
7732:
7544:
7542:U+2295
7539:) and
7527:
7525:U+22BB
7118:binary
7106:Nimber
7069:modulo
7058:binary
7000:
6994:
6964:
6958:
6914:
6908:
6887:
6881:
6860:
6848:
6208:
6202:
6172:
6166:
6122:
6116:
6072:
6066:
6045:
6039:
6018:
6012:
5872:
5869:
5866:
5860:
5857:
5854:
5766:
5763:
5760:
5754:
5751:
5748:
5715:
5694:
5691:
5688:
5682:
5679:
5676:
5562:
5559:
5556:
5550:
5547:
5544:
5520:
5222:Python
5176:, the
5064:(as a
3680:) and
2015:, and
1733:It is
1342:
1334:
418:Affine
107:
100:
93:
86:
78:
8715:False
8222:(PDF)
8145:[
8004:S2CID
7974:(PDF)
7706:(PDF)
7666:"XOR"
7652:Notes
7500:LaTeX
7498:) in
7350:adder
7210:carry
5988:GF(2)
5985:field
5505:: yes
5373:: yes
5328:, or
5178:caret
4243:as a
3801:group
1707:is a
1703:, or
112:JSTOR
98:books
8324:True
8251:2017
8229:2013
8119:ISBN
8060:2023
7759:ISBN
7730:ISBN
7683:2015
7478:and
7439:and
7411:and
7402:RAID
7393:for
7203:0111
7196:1001
7194:XOR
7189:1110
6816:Non-
6297:: no
5997:: no
5251:and
5227:The
5220:and
5214:Ruby
5210:Perl
5206:Java
3881:and
3795:are
3751:and
3134:or:
2613:and
2062:The
1884:EXOR
1338:XNOR
1320:XNOR
790:NAND
178:0110
84:news
8031:doi
7994:hdl
7986:doi
7722:doi
7533:XOR
7470:In
7371:or
7363:In
7286:odd
7116:in
7112:of
7071:2.
6820:: 0
5218:PHP
5194:C++
4910:vel
4908:to
4886:aut
4626:is
4247:in
4212:mod
4155:mod
3857:an
2280:or
2066:of
1880:EOR
1878:),
1855:or
1762:XOR
1717:odd
1653:of
1362:XOR
1104:NOT
1002:NOR
480:AND
422:yes
394:yes
149:XOR
67:by
8768::
8199:.
8175:.
8167:;
8113:.
8079:35
8077:.
8025:.
8002:.
7992:.
7982:57
7980:.
7976:.
7826:^
7773:^
7753:.
7728:.
7691:^
7674:.
7668:.
7573:•
7560:.
7271:.
7201:=
5909:.
5297:,
5216:,
5212:,
5208:,
5204:,
5200:,
5198:C#
5196:,
5068:,
4762:,
4758:,
4349:.
4280:.
4098::
3855:is
3807:.
3243:.
2035:.
1995:,
1975:,
1969:,
1942:,
1915:,
1886:,
1882:,
1871:ɔː
1845:ɔː
1829:,
1822:ɔː
1798:,
1788:ɔː
1719:.
1699:,
1695:,
1691:,
1484:,
1458:,
1398:,
1289:,
1263:,
1237:,
1208:OR
1180:,
1153:,
1130:,
1064:,
1038:,
971:,
945:,
878:,
852:,
826:,
759:,
733:,
673:,
647:,
584:,
558:,
535:,
509:,
430:no
412:no
402:no
8713:/
8579:)
8322:/
8300:e
8293:t
8286:v
8253:.
8231:.
8205:.
8153:.
8127:.
8062:.
8037:.
8033::
8027:5
8010:.
7996::
7988::
7803:.
7767:.
7738:.
7724::
7685:.
7552:(
7535:(
7443:2
7436:2
7429:2
7422:2
7415:2
7408:2
7344:.
7320:E
7314:D
7308:C
7302:B
7296:A
7257:n
7253:)
7248:Z
7244:2
7240:/
7235:Z
7231:(
7218:n
7212:)
7205:2
7198:2
7191:2
7168:.
7154:4
7150:)
7145:Z
7141:2
7137:/
7132:Z
7128:(
6961:A
6911:B
6857:B
6851:A
6750:B
6744:A
6700:B
6694:A
6620:B
6614:A
6570:B
6564:A
6432:)
6429:C
6423:B
6420:(
6378:)
6375:C
6369:A
6366:(
6318:B
6312:A
6169:A
6119:0
6069:A
6015:A
5971:)
5968:B
5962:C
5959:(
5953:)
5950:A
5944:C
5941:(
5938:=
5935:)
5932:B
5926:A
5923:(
5917:C
5901::
5718:C
5655:)
5652:B
5646:A
5643:(
5595:)
5592:C
5586:B
5583:(
5523:A
5444:A
5438:B
5394:B
5388:A
5359:.
5347:T
5336:S
5316:T
5305:S
5285:T
5279:S
5259:T
5239:S
5224:.
5202:D
5190:C
5174:^
5158:J
5138:J
5110:J
5076:J
5052:J
5000:+
4852:a
4846:b
4838:b
4832:a
4829:=
4826:b
4820:a
4738:=
4634:B
4614:A
4594:B
4574:A
4554:B
4538:A
4465:+
4445:y
4442:+
4439:x
4419:y
4416:,
4413:x
4393:+
4369:+
4262:2
4257:F
4219:)
4216:2
4209:(
4204:q
4201:+
4198:p
4195:=
4192:r
4182:q
4176:p
4173:=
4170:r
4162:)
4159:2
4152:(
4147:q
4141:p
4138:=
4135:r
4125:q
4119:p
4116:=
4113:r
4084:2
4079:F
4055:2
4050:F
4028:T
4008:F
3985:)
3979:,
3973:(
3950:2
3945:F
3921:}
3918:F
3915:,
3912:T
3909:{
3842:)
3836:,
3833:}
3830:F
3827:,
3824:T
3821:{
3818:(
3783:)
3777:,
3774:}
3771:F
3768:,
3765:T
3762:{
3759:(
3739:)
3733:,
3730:}
3727:F
3724:,
3721:T
3718:{
3715:(
3700:(
3676:(
3634:.
3631:q
3622:p
3616:q
3610:p
3601:)
3598:q
3592:p
3589:(
3550:)
3541:q
3538:p
3532:(
3529:)
3526:q
3523:+
3520:p
3517:(
3512:=
3507:)
3504:q
3498:p
3495:(
3482:)
3479:q
3473:p
3470:(
3465:=
3457:)
3449:q
3444:+
3436:p
3431:(
3428:)
3425:q
3422:+
3419:p
3416:(
3411:=
3406:)
3403:q
3394:p
3388:(
3378:)
3375:q
3369:p
3366:(
3361:=
3353:q
3345:p
3340:+
3332:q
3327:p
3322:=
3317:)
3314:q
3308:p
3302:(
3292:)
3289:q
3280:p
3277:(
3272:=
3267:q
3261:p
3201:)
3198:q
3189:p
3183:(
3177:)
3174:q
3168:p
3165:(
3160:=
3155:q
3149:p
3115:)
3112:)
3109:q
3100:p
3094:(
3088:)
3085:q
3079:p
3076:(
3073:(
3065:=
3060:q
3054:p
3027:q
3021:p
2994:)
2991:q
2985:p
2982:(
2972:)
2969:q
2963:p
2960:(
2952:=
2944:)
2941:q
2935:p
2932:(
2922:)
2919:q
2910:p
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2891:)
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2867:)
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2858:p
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2852:(
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2815:)
2812:p
2803:p
2800:(
2797:(
2792:=
2784:)
2781:q
2775:)
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2763:p
2760:(
2757:(
2747:)
2744:p
2735:)
2732:q
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