3100:
2877:
5250:
2894:
2906:
2833:
2865:
3101:
2905:
2849:
1074:
1062:
2749:
2812:
931:
970:
883:
990:
848:
828:
808:
788:
768:
748:
725:
3629:
1747:
Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has a right inverse is equivalent to the
2893:
2876:
4304:
1483:, if defined with the set of real numbers as the domain and the codomain, is not surjective (as its range is the set of positive real numbers).
640:
who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word
3329:
1583:. This is, the function together with its codomain. Unlike injectivity, surjectivity cannot be read off of the graph of the function alone.
4387:
3528:
2832:
2864:
2489:
Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijection defined on a
1532:). Under this definition, the matrix exponential is surjective for complex matrices, although still not surjective for real matrices.
1479:
is a surjective and even bijective (mapping from the set of positive real numbers to the set of all real numbers). Its inverse, the
4701:
999:
5279:
4859:
3489:
3438:
3339:
237:
3647:
2675:
4714:
4037:
3207:
2815:
447:
3424:
2754:
4719:
4709:
4446:
4299:
3652:
192:
3643:
4855:
4197:
2261:
4952:
4696:
3521:
904:
5284:
5274:
4257:
3950:
2848:
1330:= 0, and every cubic polynomial with real coefficients has at least one real root. However, this function is not
677:
of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.
65:
3691:
5213:
4915:
4678:
4673:
4498:
3919:
3603:
423:
222:
1986:
Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse
5208:
4991:
4908:
4621:
4552:
4429:
3671:
943:
5133:
4959:
4645:
4279:
3878:
2493:
of its domain by collapsing all arguments mapping to a given fixed image. More precisely, every surjection
1606:
666:
3142:= range of function. Every element in the range is mapped onto from an element in the domain, by the rule
5289:
5011:
5006:
4616:
4355:
4284:
3613:
3514:
177:
686:
4940:
4530:
3924:
3892:
3583:
2074:
of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If
257:
17:
3259:
5230:
5179:
5076:
4574:
4535:
4012:
3657:
3237:
1995:
662:
345:
3686:
5071:
5001:
4540:
4392:
4375:
4098:
3578:
2147:
1555:
In a 3D video game, vectors are projected onto a 2D flat screen by means of a surjective function.
4903:
4880:
4841:
4727:
4668:
4314:
4234:
4078:
4022:
3635:
3473:
1536:
440:
207:
132:
87:
2324:, the function applied first, need not be). These properties generalize from surjections in the
5193:
4920:
4898:
4865:
4758:
4604:
4589:
4562:
4513:
4397:
4332:
4157:
4123:
4118:
3992:
3823:
3800:
3146:. There may be a number of domain elements which map to the same range element. That is, every
2333:
1952:
696:
505:
40:
5123:
4976:
4768:
4486:
4222:
4128:
3987:
3972:
3853:
3828:
2027:
403:
3467:
3286:
853:
5096:
5058:
4935:
4739:
4579:
4503:
4481:
4309:
4267:
4166:
4133:
3997:
3785:
3696:
2914:
in the
Cartesian plane. Although some parts of the function are surjective, where elements
2520:
2273:
2059:
1944:
1666:
1579:, then surjectivity is not a property of the function itself, but rather a property of the
1510:
1498:
1480:
1081:
728:
674:
588:
534:
388:
350:
162:
73:
8:
5225:
5116:
5101:
5081:
5038:
4925:
4875:
4801:
4746:
4683:
4476:
4471:
4419:
4176:
3848:
3748:
3676:
3667:
3663:
3598:
3593:
2023:
1576:
1518:
1103:
1099:
700:
655:
572:
312:
307:
5254:
5023:
4986:
4971:
4964:
4947:
4751:
4733:
4599:
4525:
4508:
4461:
4274:
4183:
4017:
4002:
3962:
3914:
3899:
3887:
3843:
3818:
3588:
3537:
3430:
2439:
2340:
1890:
1569:
1565:
1487:
1335:
1331:
1244:
1199:
975:
833:
813:
793:
773:
753:
733:
710:
624:
618:
433:
317:
292:
4207:
5249:
5189:
4996:
4806:
4796:
4688:
4569:
4404:
4380:
4161:
4145:
4050:
4027:
3904:
3873:
3838:
3733:
3568:
3495:
3485:
3434:
3335:
2512:
2003:
1689:
1580:
1540:
1529:
1469:
1159:
393:
297:
287:
272:
267:
5203:
5198:
5091:
5048:
4870:
4831:
4826:
4811:
4637:
4594:
4491:
4289:
4239:
3813:
3775:
3477:
3463:
2325:
1960:
1705:
629:
478:
383:
360:
117:
658:
of the domain of a surjective function completely covers the function's codomain.
5184:
5174:
5128:
5111:
5066:
5028:
4930:
4850:
4657:
4584:
4557:
4545:
4451:
4365:
4339:
4294:
4262:
4063:
3865:
3808:
3758:
3723:
3681:
3212:
2101:
2097:
1748:
670:
378:
355:
302:
1114:
surjective, because the image does not fill the whole codomain. In other words,
5169:
5148:
5106:
5086:
4981:
4836:
4434:
4424:
4414:
4409:
4343:
4217:
4093:
3982:
3977:
3955:
3556:
3381:
3378:
3355:
2669:
2568:
2039:
398:
3481:
5268:
5143:
4821:
4328:
4113:
4103:
4073:
4058:
3728:
2435:
637:
5043:
4890:
4791:
4783:
4663:
4611:
4520:
4456:
4439:
4370:
4229:
4088:
3790:
3573:
3227:
3217:
2490:
1959:. Specifically, surjective functions are precisely the epimorphisms in the
673:, and every function with a right inverse is necessarily a surjection. The
102:
3287:"Bijection, Injection, And Surjection | Brilliant Math & Science Wiki"
1501:
to itself. It is, however, usually defined as a map from the space of all
31:
5153:
5033:
4212:
4202:
4149:
3833:
3753:
3738:
3618:
3563:
3222:
2665:
2329:
2071:
1956:
1312:
1248:
1204:
665:
its codomain to the image of its domain. Every surjective function has a
642:
461:
2507:
can be factored as a projection followed by a bijection as follows. Let
4083:
3938:
3909:
3715:
2170:
1211:
282:
277:
1552:
to one of its factors is surjective, unless the other factor is empty.
885:. Surjections are sometimes denoted by a two-headed rightwards arrow (
5235:
5138:
4191:
4108:
4068:
4032:
3968:
3780:
3770:
3743:
3506:
3232:
2192:
327:
5220:
5018:
4466:
4171:
3765:
3311:
2406:
1948:
1803:
1088:
704:
520:
408:
147:
77:
27:
Mathematical function such that every output has at least one input
3499:
2899:
Surjective composition: the first function need not be surjective.
4816:
3608:
2870:
An injective non-surjective function (injection, not a bijection)
1207:
888:
413:
1955:
and their composition. Right-cancellative morphisms are called
1773:
633:
1943:. This property is formulated in terms of functions and their
4360:
3706:
3551:
2882:
A non-injective non-surjective function (neither a bijection)
496:
2065:
1586:
1073:
194:
2484:
1490:
is not surjective when seen as a map from the space of all
1134:) is colored yellow; Second, all the rest of the points in
1057:{\displaystyle \forall y\in Y,\,\exists x\in X,\;\;f(x)=y}
484:
1947:
and can be generalized to the more general notion of the
1350:= 2}. (In fact, the pre-image of this function for every
2339:
Any function can be decomposed into a surjection and an
1575:
If (as is often done) a function is identified with its
2744:{\textstyle |B|!{\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}
2763:
2757:
2700:
2678:
1142:
would be surjective only if there were no blue points.
1138:, that are not yellow, are colored blue. The function
3310:
Miller, Jeff, "Injection, Surjection and
Bijection",
1318:
is the solution set of the cubic polynomial equation
1002:
978:
946:
907:
856:
836:
816:
796:
776:
756:
736:
713:
487:
493:
2807:{\textstyle {\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}
2668:of this set is one of the twelve aspects of Rota's
1118:is colored in a two-step process: First, for every
481:
123:
2806:
2743:
2009:
1056:
984:
964:
925:
877:
842:
822:
802:
782:
762:
742:
719:
3313:Earliest Uses of Some of the Words of Mathematics
2276:of surjective functions is always surjective: If
1724:, may not be the identity function on the domain
119:
5266:
2477:is surjective since it is a projection map, and
2267:
239:
3110:in the Cartesian plane, defined by the mapping
2228:. Using the axiom of choice one can show that
2002:. A morphism with a right inverse is called a
1870:
1819:For example, in the first illustration in the
3522:
3407:
3348:
2058:that is right-unique and both left-total and
926:{\displaystyle f\colon X\twoheadrightarrow Y}
441:
2220:is empty or that there is a surjection from
1744:, but cannot necessarily be reversed by it.
1712:because the composition in the other order,
1311:is surjective, because the pre-image of any
95:
490:
30:"Onto" redirects here. For other uses, see
3714:
3529:
3515:
3334:. American Mathematical Soc. p. 106.
1393:surjective, since there is no real number
1035:
1034:
448:
434:
104:
3422:
3382:"Injections, Surjections, and Bijections"
2641:
2284:are both surjective, and the codomain of
2146:is easily seen to be injective, thus the
2066:Cardinality of the domain of a surjection
1587:Surjections as right invertible functions
1568:if and only if it is both surjective and
1018:
2485:Induced surjection and induced bijection
1338:), since, for example, the pre-image of
1072:
3426:Topoi, the Categorial Analysis of Logic
3327:
2173:with the same number of elements, then
224:
215:
14:
5267:
3536:
2842:function (surjection, not a bijection)
2654:, one can form the set of surjections
2600:be the well-defined function given by
1967:is derived from the Greek preposition
965:{\displaystyle f\colon X\rightarrow Y}
687:Function (mathematics) § Notation
3510:
3260:"Injective, Surjective and Bijective"
3174:Only one domain is shown which makes
2551:/~ is the set of all preimages under
1851:is not unique (it would also work if
685:Further information on notation:
661:Any function induces a surjection by
243:
228:
209:
198:
183:
164:
153:
134:
108:
89:
3281:
3279:
3254:
3252:
2042:. A surjective function with domain
179:
170:
3208:Bijection, injection and surjection
1889:is surjective if and only if it is
1835:) = 4. There is also some function
654:, and relates to the fact that the
24:
3456:
3377:
3309:
2816:Stirling number of the second kind
2050:is then a binary relation between
1859:) equals 3); it only matters that
1019:
1003:
25:
5301:
3276:
3249:
2092:has at least as many elements as
1443:in the nonnegative real codomain
1094:. The smaller yellow oval inside
551:. In other words, for a function
5248:
3099:
2904:
2892:
2875:
2863:
2847:
2831:
597:may map one or more elements of
477:
149:
140:
2088:is a surjective function, then
2010:Surjections as binary relations
1435:(with the restricted codomain)
1358:≤ 2 has more than one element.)
66:History of the function concept
3416:
3401:
3371:
3321:
3303:
2792:
2784:
2775:
2767:
2729:
2721:
2712:
2704:
2688:
2680:
2304:is surjective. Conversely, if
2100:. (The proof appeals to the
1045:
1039:
956:
917:
866:
860:
13:
1:
5209:History of mathematical logic
3462:
3412:. Addison-Wesley. p. 35.
3243:
2268:Composition and decomposition
2187:is surjective if and only if
1559:
1214:integers to 1) is surjective.
680:
513:such that, for every element
5280:Basic concepts in set theory
5134:Primitive recursive function
2481:is injective by definition.
1820:
1439:surjective, since for every
1243:+ 1 is surjective (and even
1147:
707:. Equivalently, a function
7:
3423:Goldblatt, Robert (2006) .
3201:
2216:is used to say that either
2038:by identifying it with its
1871:Surjections as epimorphisms
1068:
992:is said to be surjective if
899:RIGHTWARDS TWO HEADED ARROW
770:is surjective if for every
10:
5306:
4198:Schröder–Bernstein theorem
3925:Monadic predicate calculus
3584:Foundations of mathematics
3154:is mapped from an element
2821:
2288:is equal to the domain of
2262:Schröder–Bernstein theorem
1847:. It doesn't matter that
1802:can be recovered from its
1145:
810:there exists at least one
684:
581:. It is not required that
424:List of specific functions
29:
5244:
5231:Philosophy of mathematics
5180:Automated theorem proving
5162:
5057:
4889:
4782:
4634:
4351:
4327:
4305:Von Neumann–Bernays–Gödel
4250:
4144:
4048:
3946:
3937:
3864:
3799:
3705:
3627:
3544:
3482:10.1007/978-3-642-59309-3
3476:. Vol. 1. Springer.
3328:Mashaal, Maurice (2006).
3238:Section (category theory)
2579:to its equivalence class
2357:there exist a surjection
2014:Any function with domain
1823:, there is some function
1078:A non-surjective function
575:of the function's domain
2912:Non-surjective functions
2461:carries each element of
2104:to show that a function
2030:binary relation between
1447:, there is at least one
1404:. However, the function
4881:Self-verifying theories
4702:Tarski's axiomatization
3653:Tarski's undefinability
3648:incompleteness theorems
3474:Elements of Mathematics
2942:), some parts are not.
2473:sends its points. Then
2457:which contains it, and
1704:need not be a complete
1146:For more examples, see
603:to the same element of
5285:Mathematical relations
5275:Functions and mappings
5255:Mathematics portal
4866:Proof of impossibility
4514:propositional variable
3824:Propositional calculus
3408:T. M. Apostol (1981).
3138:= domain of function,
2808:
2745:
2642:The set of surjections
2434:. These preimages are
2401:. To see this, define
2161:Specifically, if both
1893:: given any functions
1661:is a right inverse of
1270:: such an appropriate
1143:
1058:
986:
966:
927:
879:
878:{\displaystyle f(x)=y}
844:
824:
804:
784:
764:
744:
721:
616:and the related terms
5124:Kolmogorov complexity
5077:Computably enumerable
4977:Model complete theory
4769:Principia Mathematica
3829:Propositional formula
3658:Banach–Tarski paradox
3410:Mathematical Analysis
3182:two possible domains
2809:
2746:
1688:in that order is the
1247:), because for every
1076:
1059:
987:
967:
928:
880:
845:
825:
805:
785:
765:
745:
722:
5072:Church–Turing thesis
5059:Computability theory
4268:continuum hypothesis
3786:Square of opposition
3644:Gödel's completeness
3429:(Revised ed.).
3166:can map to the same
3108:surjective functions
2858:function (bijection)
2755:
2676:
2521:equivalence relation
2519:under the following
2316:is surjective, then
2248:together imply that
1657:). In other words,
1511:general linear group
1481:exponential function
1474:ln : (0, +∞) →
1210:are mapped to 0 and
1000:
976:
944:
905:
854:
834:
814:
794:
774:
754:
734:
711:
632:, a group of mainly
5226:Mathematical object
5117:P versus NP problem
5082:Computable function
4876:Reverse mathematics
4802:Logical consequence
4679:primitive recursive
4674:elementary function
4447:Free/bound variable
4300:Tarski–Grothendieck
3819:Logical connectives
3749:Logical equivalence
3599:Logical consequence
3106:Interpretation for
3015:, but there are no
2513:equivalence classes
2343:: For any function
2320:is surjective (but
1530:invertible matrices
1451:in the real domain
1110:. This function is
693:surjective function
628:were introduced by
466:surjective function
5290:Types of functions
5024:Transfer principle
4987:Semantics of logic
4972:Categorical theory
4948:Non-standard model
4462:Logical connective
3589:Information theory
3538:Mathematical logic
3431:Dover Publications
3356:"Arrows – Unicode"
3264:www.mathsisfun.com
2957:, but there is no
2804:
2798:
2741:
2735:
2672:, and is given by
2453:to the element of
2096:, in the sense of
1891:right-cancellative
1768:is surjective and
1732:. In other words,
1488:matrix exponential
1144:
1054:
982:
962:
923:
875:
840:
820:
800:
780:
760:
740:
717:
533:in the function's
519:of the function's
258:Classes/properties
5262:
5261:
5194:Abstract category
4997:Theories of truth
4807:Rule of inference
4797:Natural deduction
4778:
4777:
4323:
4322:
4028:Cartesian product
3933:
3932:
3839:Many-valued logic
3814:Boolean functions
3697:Russell's paradox
3672:diagonal argument
3569:First-order logic
3491:978-3-540-22525-6
3440:978-0-486-45026-1
3341:978-0-8218-3967-6
2571:which sends each
2547:). Equivalently,
2405:to be the set of
2396:
2371:and an injection
2311:
2299:
2260:a variant of the
2158:| is satisfied.)
2148:formal definition
2022:can be seen as a
2004:split epimorphism
1928:
1918:
1719:
1690:identity function
1675:
1653:can be undone by
1541:cartesian product
1470:natural logarithm
1160:identity function
985:{\displaystyle f}
843:{\displaystyle X}
823:{\displaystyle x}
803:{\displaystyle Y}
783:{\displaystyle y}
763:{\displaystyle Y}
743:{\displaystyle X}
720:{\displaystyle f}
458:
457:
370:Generalizations
16:(Redirected from
5297:
5253:
5252:
5204:History of logic
5199:Category of sets
5092:Decision problem
4871:Ordinal analysis
4812:Sequent calculus
4710:Boolean algebras
4650:
4649:
4624:
4595:logical/constant
4349:
4348:
4335:
4258:Zermelo–Fraenkel
4009:Set operations:
3944:
3943:
3881:
3712:
3711:
3692:Löwenheim–Skolem
3579:Formal semantics
3531:
3524:
3517:
3508:
3507:
3503:
3451:
3450:
3448:
3447:
3420:
3414:
3413:
3405:
3399:
3398:
3396:
3395:
3386:
3375:
3369:
3368:
3366:
3365:
3360:
3352:
3346:
3345:
3325:
3319:
3317:
3307:
3301:
3300:
3298:
3297:
3283:
3274:
3273:
3271:
3270:
3256:
3162:, more than one
3103:
2922:do have a value
2908:
2896:
2879:
2867:
2851:
2838:A non-injective
2835:
2813:
2811:
2810:
2805:
2803:
2802:
2795:
2787:
2778:
2770:
2750:
2748:
2747:
2742:
2740:
2739:
2732:
2724:
2715:
2707:
2691:
2683:
2663:
2653:
2649:
2506:
2465:to the point in
2433:
2418:
2400:
2394:
2384:
2370:
2356:
2326:category of sets
2315:
2309:
2303:
2297:
2259:
2247:
2237:
2215:
2186:
2117:
2098:cardinal numbers
2087:
1961:category of sets
1942:
1932:
1926:
1916:
1910:
1888:
1815:
1797:
1767:
1723:
1717:
1679:
1673:
1640:
1622:
1609:of the function
1605:is said to be a
1604:
1551:
1509:matrices to the
1478:
1464:
1434:
1420:
1403:
1388:
1374:
1294:
1230:
1185:
1063:
1061:
1060:
1055:
991:
989:
988:
983:
971:
969:
968:
963:
932:
930:
929:
924:
900:
897:
894:
892:
884:
882:
881:
876:
849:
847:
846:
841:
829:
827:
826:
821:
809:
807:
806:
801:
789:
787:
786:
781:
769:
767:
766:
761:
749:
747:
746:
741:
726:
724:
723:
718:
703:is equal to its
630:Nicolas Bourbaki
608:
602:
596:
586:
580:
570:
564:
550:
532:
518:
512:
503:
502:
499:
498:
495:
492:
489:
486:
483:
450:
443:
436:
248:
247:
241:
233:
232:
226:
218:
217:
213:
203:
202:
196:
188:
187:
181:
173:
172:
168:
158:
157:
151:
143:
142:
138:
128:
127:
121:
113:
112:
106:
98:
97:
93:
60:
37:
36:
21:
5305:
5304:
5300:
5299:
5298:
5296:
5295:
5294:
5265:
5264:
5263:
5258:
5247:
5240:
5185:Category theory
5175:Algebraic logic
5158:
5129:Lambda calculus
5067:Church encoding
5053:
5029:Truth predicate
4885:
4851:Complete theory
4774:
4643:
4639:
4635:
4630:
4622:
4342: and
4338:
4333:
4319:
4295:New Foundations
4263:axiom of choice
4246:
4208:Gödel numbering
4148: and
4140:
4044:
3929:
3879:
3860:
3809:Boolean algebra
3795:
3759:Equiconsistency
3724:Classical logic
3701:
3682:Halting problem
3670: and
3646: and
3634: and
3633:
3628:Theorems (
3623:
3540:
3535:
3492:
3459:
3457:Further reading
3454:
3445:
3443:
3441:
3421:
3417:
3406:
3402:
3393:
3391:
3389:math.umaine.edu
3384:
3376:
3372:
3363:
3361:
3358:
3354:
3353:
3349:
3342:
3326:
3322:
3308:
3304:
3295:
3293:
3285:
3284:
3277:
3268:
3266:
3258:
3257:
3250:
3246:
3213:Cover (algebra)
3204:
3197:
3195:
3188:
3104:
3095:
3093:
3082:
3075:
3064:
3057:
3046:
3035:
3028:
3021:
3010:
3003:
2996:
2985:
2974:
2963:
2952:
2909:
2900:
2897:
2888:
2887:
2886:
2883:
2880:
2871:
2868:
2859:
2852:
2843:
2836:
2824:
2797:
2796:
2791:
2783:
2780:
2779:
2774:
2766:
2759:
2758:
2756:
2753:
2752:
2734:
2733:
2728:
2720:
2717:
2716:
2711:
2703:
2696:
2695:
2687:
2679:
2677:
2674:
2673:
2655:
2651:
2647:
2644:
2633:
2612:
2608:
2591:
2582:
2531:if and only if
2494:
2487:
2424:
2409:
2386:
2372:
2358:
2344:
2305:
2293:
2270:
2249:
2239:
2229:
2207:
2206:, the notation
2198:Given two sets
2174:
2105:
2102:axiom of choice
2075:
2068:
2012:
1934:
1912:
1894:
1876:
1873:
1806:
1781:
1755:
1749:axiom of choice
1713:
1700:. The function
1669:
1624:
1610:
1592:
1589:
1562:
1543:
1473:
1456:
1422:
1419:
1405:
1398:
1376:
1362:
1334:(and hence not
1282:
1218:
1176:
1167:
1151:
1071:
1001:
998:
997:
977:
974:
973:
945:
942:
941:
906:
903:
902:
898:
895:
887:
886:
855:
852:
851:
835:
832:
831:
815:
812:
811:
795:
792:
791:
775:
772:
771:
755:
752:
751:
735:
732:
731:
712:
709:
708:
689:
683:
671:axiom of choice
604:
598:
592:
591:; the function
582:
576:
566:
565:, the codomain
552:
538:
528:
523:, there exists
514:
508:
480:
476:
468:(also known as
454:
418:
379:Binary relation
365:
332:
252:
246:
238:
231:
223:
212:
208:
201:
193:
186:
178:
167:
163:
156:
148:
137:
133:
126:
118:
111:
103:
92:
88:
47:
35:
32:wiktionary:onto
28:
23:
22:
15:
12:
11:
5:
5303:
5293:
5292:
5287:
5282:
5277:
5260:
5259:
5245:
5242:
5241:
5239:
5238:
5233:
5228:
5223:
5218:
5217:
5216:
5206:
5201:
5196:
5187:
5182:
5177:
5172:
5170:Abstract logic
5166:
5164:
5160:
5159:
5157:
5156:
5151:
5149:Turing machine
5146:
5141:
5136:
5131:
5126:
5121:
5120:
5119:
5114:
5109:
5104:
5099:
5089:
5087:Computable set
5084:
5079:
5074:
5069:
5063:
5061:
5055:
5054:
5052:
5051:
5046:
5041:
5036:
5031:
5026:
5021:
5016:
5015:
5014:
5009:
5004:
4994:
4989:
4984:
4982:Satisfiability
4979:
4974:
4969:
4968:
4967:
4957:
4956:
4955:
4945:
4944:
4943:
4938:
4933:
4928:
4923:
4913:
4912:
4911:
4906:
4899:Interpretation
4895:
4893:
4887:
4886:
4884:
4883:
4878:
4873:
4868:
4863:
4853:
4848:
4847:
4846:
4845:
4844:
4834:
4829:
4819:
4814:
4809:
4804:
4799:
4794:
4788:
4786:
4780:
4779:
4776:
4775:
4773:
4772:
4764:
4763:
4762:
4761:
4756:
4755:
4754:
4749:
4744:
4724:
4723:
4722:
4720:minimal axioms
4717:
4706:
4705:
4704:
4693:
4692:
4691:
4686:
4681:
4676:
4671:
4666:
4653:
4651:
4632:
4631:
4629:
4628:
4627:
4626:
4614:
4609:
4608:
4607:
4602:
4597:
4592:
4582:
4577:
4572:
4567:
4566:
4565:
4560:
4550:
4549:
4548:
4543:
4538:
4533:
4523:
4518:
4517:
4516:
4511:
4506:
4496:
4495:
4494:
4489:
4484:
4479:
4474:
4469:
4459:
4454:
4449:
4444:
4443:
4442:
4437:
4432:
4427:
4417:
4412:
4410:Formation rule
4407:
4402:
4401:
4400:
4395:
4385:
4384:
4383:
4373:
4368:
4363:
4358:
4352:
4346:
4329:Formal systems
4325:
4324:
4321:
4320:
4318:
4317:
4312:
4307:
4302:
4297:
4292:
4287:
4282:
4277:
4272:
4271:
4270:
4265:
4254:
4252:
4248:
4247:
4245:
4244:
4243:
4242:
4232:
4227:
4226:
4225:
4218:Large cardinal
4215:
4210:
4205:
4200:
4195:
4181:
4180:
4179:
4174:
4169:
4154:
4152:
4142:
4141:
4139:
4138:
4137:
4136:
4131:
4126:
4116:
4111:
4106:
4101:
4096:
4091:
4086:
4081:
4076:
4071:
4066:
4061:
4055:
4053:
4046:
4045:
4043:
4042:
4041:
4040:
4035:
4030:
4025:
4020:
4015:
4007:
4006:
4005:
4000:
3990:
3985:
3983:Extensionality
3980:
3978:Ordinal number
3975:
3965:
3960:
3959:
3958:
3947:
3941:
3935:
3934:
3931:
3930:
3928:
3927:
3922:
3917:
3912:
3907:
3902:
3897:
3896:
3895:
3885:
3884:
3883:
3870:
3868:
3862:
3861:
3859:
3858:
3857:
3856:
3851:
3846:
3836:
3831:
3826:
3821:
3816:
3811:
3805:
3803:
3797:
3796:
3794:
3793:
3788:
3783:
3778:
3773:
3768:
3763:
3762:
3761:
3751:
3746:
3741:
3736:
3731:
3726:
3720:
3718:
3709:
3703:
3702:
3700:
3699:
3694:
3689:
3684:
3679:
3674:
3662:Cantor's
3660:
3655:
3650:
3640:
3638:
3625:
3624:
3622:
3621:
3616:
3611:
3606:
3601:
3596:
3591:
3586:
3581:
3576:
3571:
3566:
3561:
3560:
3559:
3548:
3546:
3542:
3541:
3534:
3533:
3526:
3519:
3511:
3505:
3504:
3490:
3469:Theory of Sets
3458:
3455:
3453:
3452:
3439:
3415:
3400:
3370:
3347:
3340:
3320:
3302:
3275:
3247:
3245:
3242:
3241:
3240:
3235:
3230:
3225:
3220:
3215:
3210:
3203:
3200:
3199:
3198:
3193:
3186:
3105:
3098:
3096:
3091:
3080:
3073:
3062:
3055:
3044:
3033:
3026:
3019:
3008:
3001:
2994:
2983:
2972:
2961:
2950:
2910:
2903:
2901:
2898:
2891:
2885:
2884:
2881:
2874:
2872:
2869:
2862:
2860:
2853:
2846:
2844:
2837:
2830:
2827:
2826:
2825:
2823:
2820:
2801:
2794:
2790:
2786:
2782:
2781:
2777:
2773:
2769:
2765:
2764:
2762:
2738:
2731:
2727:
2723:
2719:
2718:
2714:
2710:
2706:
2702:
2701:
2699:
2694:
2690:
2686:
2682:
2670:Twelvefold way
2643:
2640:
2629:
2610:
2604:
2587:
2580:
2569:projection map
2486:
2483:
2269:
2266:
2067:
2064:
2040:function graph
2011:
2008:
1990:of a morphism
1872:
1869:
1692:on the domain
1588:
1585:
1564:A function is
1561:
1558:
1557:
1556:
1553:
1533:
1517:(that is, the
1484:
1466:
1417:
1359:
1279:
1215:
1173:
1172:is surjective.
1163:
1148:§ Gallery
1070:
1067:
1066:
1065:
1053:
1050:
1047:
1044:
1041:
1038:
1033:
1030:
1027:
1024:
1021:
1017:
1014:
1011:
1008:
1005:
994:
993:
981:
961:
958:
955:
952:
949:
936:Symbolically,
922:
919:
916:
913:
910:
874:
871:
868:
865:
862:
859:
839:
819:
799:
779:
759:
739:
716:
682:
679:
638:mathematicians
526:
456:
455:
453:
452:
445:
438:
430:
427:
426:
420:
419:
417:
416:
411:
406:
401:
396:
391:
386:
381:
375:
372:
371:
367:
366:
364:
363:
358:
353:
348:
342:
339:
338:
334:
333:
331:
330:
325:
320:
315:
310:
305:
300:
295:
290:
285:
280:
275:
270:
264:
261:
260:
254:
253:
251:
250:
244:
235:
229:
220:
210:
205:
199:
190:
184:
175:
165:
160:
154:
145:
135:
130:
124:
115:
109:
100:
90:
84:
81:
80:
69:
68:
62:
61:
44:
43:
26:
9:
6:
4:
3:
2:
5302:
5291:
5288:
5286:
5283:
5281:
5278:
5276:
5273:
5272:
5270:
5257:
5256:
5251:
5243:
5237:
5234:
5232:
5229:
5227:
5224:
5222:
5219:
5215:
5212:
5211:
5210:
5207:
5205:
5202:
5200:
5197:
5195:
5191:
5188:
5186:
5183:
5181:
5178:
5176:
5173:
5171:
5168:
5167:
5165:
5161:
5155:
5152:
5150:
5147:
5145:
5144:Recursive set
5142:
5140:
5137:
5135:
5132:
5130:
5127:
5125:
5122:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5094:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5073:
5070:
5068:
5065:
5064:
5062:
5060:
5056:
5050:
5047:
5045:
5042:
5040:
5037:
5035:
5032:
5030:
5027:
5025:
5022:
5020:
5017:
5013:
5010:
5008:
5005:
5003:
5000:
4999:
4998:
4995:
4993:
4990:
4988:
4985:
4983:
4980:
4978:
4975:
4973:
4970:
4966:
4963:
4962:
4961:
4958:
4954:
4953:of arithmetic
4951:
4950:
4949:
4946:
4942:
4939:
4937:
4934:
4932:
4929:
4927:
4924:
4922:
4919:
4918:
4917:
4914:
4910:
4907:
4905:
4902:
4901:
4900:
4897:
4896:
4894:
4892:
4888:
4882:
4879:
4877:
4874:
4872:
4869:
4867:
4864:
4861:
4860:from ZFC
4857:
4854:
4852:
4849:
4843:
4840:
4839:
4838:
4835:
4833:
4830:
4828:
4825:
4824:
4823:
4820:
4818:
4815:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4793:
4790:
4789:
4787:
4785:
4781:
4771:
4770:
4766:
4765:
4760:
4759:non-Euclidean
4757:
4753:
4750:
4748:
4745:
4743:
4742:
4738:
4737:
4735:
4732:
4731:
4729:
4725:
4721:
4718:
4716:
4713:
4712:
4711:
4707:
4703:
4700:
4699:
4698:
4694:
4690:
4687:
4685:
4682:
4680:
4677:
4675:
4672:
4670:
4667:
4665:
4662:
4661:
4659:
4655:
4654:
4652:
4647:
4641:
4636:Example
4633:
4625:
4620:
4619:
4618:
4615:
4613:
4610:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4587:
4586:
4583:
4581:
4578:
4576:
4573:
4571:
4568:
4564:
4561:
4559:
4556:
4555:
4554:
4551:
4547:
4544:
4542:
4539:
4537:
4534:
4532:
4529:
4528:
4527:
4524:
4522:
4519:
4515:
4512:
4510:
4507:
4505:
4502:
4501:
4500:
4497:
4493:
4490:
4488:
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4468:
4465:
4464:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4441:
4438:
4436:
4433:
4431:
4428:
4426:
4423:
4422:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4399:
4396:
4394:
4393:by definition
4391:
4390:
4389:
4386:
4382:
4379:
4378:
4377:
4374:
4372:
4369:
4367:
4364:
4362:
4359:
4357:
4354:
4353:
4350:
4347:
4345:
4341:
4336:
4330:
4326:
4316:
4313:
4311:
4308:
4306:
4303:
4301:
4298:
4296:
4293:
4291:
4288:
4286:
4283:
4281:
4280:Kripke–Platek
4278:
4276:
4273:
4269:
4266:
4264:
4261:
4260:
4259:
4256:
4255:
4253:
4249:
4241:
4238:
4237:
4236:
4233:
4231:
4228:
4224:
4221:
4220:
4219:
4216:
4214:
4211:
4209:
4206:
4204:
4201:
4199:
4196:
4193:
4189:
4185:
4182:
4178:
4175:
4173:
4170:
4168:
4165:
4164:
4163:
4159:
4156:
4155:
4153:
4151:
4147:
4143:
4135:
4132:
4130:
4127:
4125:
4124:constructible
4122:
4121:
4120:
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4085:
4082:
4080:
4077:
4075:
4072:
4070:
4067:
4065:
4062:
4060:
4057:
4056:
4054:
4052:
4047:
4039:
4036:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4010:
4008:
4004:
4001:
3999:
3996:
3995:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3970:
3966:
3964:
3961:
3957:
3954:
3953:
3952:
3949:
3948:
3945:
3942:
3940:
3936:
3926:
3923:
3921:
3918:
3916:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3894:
3891:
3890:
3889:
3886:
3882:
3877:
3876:
3875:
3872:
3871:
3869:
3867:
3863:
3855:
3852:
3850:
3847:
3845:
3842:
3841:
3840:
3837:
3835:
3832:
3830:
3827:
3825:
3822:
3820:
3817:
3815:
3812:
3810:
3807:
3806:
3804:
3802:
3801:Propositional
3798:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3772:
3769:
3767:
3764:
3760:
3757:
3756:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3729:Logical truth
3727:
3725:
3722:
3721:
3719:
3717:
3713:
3710:
3708:
3704:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3669:
3665:
3661:
3659:
3656:
3654:
3651:
3649:
3645:
3642:
3641:
3639:
3637:
3631:
3626:
3620:
3617:
3615:
3612:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3570:
3567:
3565:
3562:
3558:
3555:
3554:
3553:
3550:
3549:
3547:
3543:
3539:
3532:
3527:
3525:
3520:
3518:
3513:
3512:
3509:
3501:
3497:
3493:
3487:
3483:
3479:
3475:
3471:
3470:
3465:
3461:
3460:
3442:
3436:
3432:
3428:
3427:
3419:
3411:
3404:
3390:
3383:
3380:
3379:Farlow, S. J.
3374:
3357:
3351:
3343:
3337:
3333:
3332:
3324:
3315:
3314:
3306:
3292:
3291:brilliant.org
3288:
3282:
3280:
3265:
3261:
3255:
3253:
3248:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3214:
3211:
3209:
3206:
3205:
3192:
3185:
3181:
3177:
3173:
3169:
3165:
3161:
3157:
3153:
3149:
3145:
3141:
3137:
3133:
3129:
3125:
3121:
3117:
3113:
3109:
3102:
3097:
3090:
3086:
3079:
3072:
3068:
3061:
3054:
3050:
3043:
3039:
3032:
3025:
3018:
3014:
3007:
3000:
2993:
2989:
2982:
2978:
2971:
2967:
2960:
2956:
2949:
2945:
2941:
2937:
2933:
2929:
2925:
2921:
2917:
2913:
2907:
2902:
2895:
2890:
2889:
2878:
2873:
2866:
2861:
2857:
2854:An injective
2850:
2845:
2841:
2834:
2829:
2828:
2819:
2817:
2799:
2788:
2771:
2760:
2736:
2725:
2708:
2697:
2692:
2684:
2671:
2667:
2662:
2658:
2639:
2637:
2632:
2628:
2624:
2620:
2616:
2607:
2603:
2599:
2595:
2590:
2586:
2578:
2574:
2570:
2566:
2562:
2558:
2554:
2550:
2546:
2542:
2538:
2534:
2530:
2526:
2522:
2518:
2514:
2510:
2505:
2501:
2497:
2492:
2482:
2480:
2476:
2472:
2468:
2464:
2460:
2456:
2452:
2449:carries each
2448:
2444:
2441:
2437:
2431:
2427:
2422:
2416:
2412:
2408:
2404:
2399:
2393:
2389:
2383:
2379:
2375:
2369:
2365:
2361:
2355:
2351:
2347:
2342:
2337:
2335:
2331:
2327:
2323:
2319:
2314:
2308:
2302:
2296:
2291:
2287:
2283:
2279:
2275:
2265:
2263:
2257:
2253:
2246:
2242:
2236:
2232:
2227:
2223:
2219:
2214:
2210:
2205:
2201:
2196:
2194:
2190:
2185:
2181:
2177:
2172:
2168:
2164:
2159:
2157:
2153:
2149:
2145:
2141:
2137:
2133:
2129:
2125:
2121:
2116:
2112:
2108:
2103:
2099:
2095:
2091:
2086:
2082:
2078:
2073:
2063:
2061:
2057:
2053:
2049:
2046:and codomain
2045:
2041:
2037:
2033:
2029:
2025:
2021:
2018:and codomain
2017:
2007:
2005:
2001:
1997:
1993:
1989:
1984:
1982:
1978:
1974:
1970:
1966:
1963:. The prefix
1962:
1958:
1954:
1950:
1946:
1941:
1937:
1931:
1925:
1921:
1915:
1909:
1905:
1901:
1897:
1892:
1887:
1883:
1879:
1868:
1866:
1862:
1858:
1854:
1850:
1846:
1842:
1838:
1834:
1830:
1826:
1822:
1817:
1813:
1809:
1805:
1801:
1796:
1792:
1788:
1784:
1779:
1775:
1771:
1766:
1762:
1758:
1752:
1750:
1745:
1743:
1739:
1736:can undo or "
1735:
1731:
1727:
1722:
1716:
1711:
1707:
1703:
1699:
1695:
1691:
1687:
1683:
1678:
1672:
1668:
1664:
1660:
1656:
1652:
1648:
1644:
1639:
1635:
1631:
1627:
1621:
1617:
1613:
1608:
1607:right inverse
1603:
1599:
1595:
1591:The function
1584:
1582:
1578:
1573:
1571:
1567:
1554:
1550:
1546:
1542:
1538:
1534:
1531:
1528:
1524:
1520:
1516:
1512:
1508:
1504:
1500:
1497:
1493:
1489:
1485:
1482:
1477:
1471:
1467:
1463:
1459:
1454:
1450:
1446:
1442:
1438:
1433:
1429:
1425:
1416:
1412:
1408:
1401:
1396:
1392:
1387:
1383:
1379:
1373:
1369:
1365:
1361:The function
1360:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1314:
1310:
1306:
1302:
1298:
1293:
1289:
1285:
1281:The function
1280:
1277:
1273:
1269:
1265:
1261:
1257:
1254:, we have an
1253:
1250:
1246:
1242:
1238:
1234:
1229:
1225:
1221:
1217:The function
1216:
1213:
1209:
1206:
1202:
1201:
1197:
1193:
1189:
1183:
1179:
1175:The function
1174:
1171:
1166:
1161:
1157:
1153:
1152:
1149:
1141:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:
1102:(also called
1101:
1097:
1093:
1090:
1086:
1083:
1079:
1075:
1051:
1048:
1042:
1036:
1031:
1028:
1025:
1022:
1015:
1012:
1009:
1006:
996:
995:
979:
959:
953:
950:
947:
939:
938:
937:
934:
920:
914:
911:
908:
890:
872:
869:
863:
857:
837:
817:
797:
777:
757:
750:and codomain
737:
730:
714:
706:
702:
698:
694:
688:
678:
676:
672:
669:assuming the
668:
667:right inverse
664:
659:
657:
653:
649:
645:
644:
639:
636:20th-century
635:
631:
627:
626:
621:
620:
615:
610:
607:
601:
595:
590:
585:
579:
574:
569:
563:
559:
555:
549:
545:
541:
536:
531:
524:
522:
517:
511:
507:
501:
475:
474:onto function
471:
467:
463:
451:
446:
444:
439:
437:
432:
431:
429:
428:
425:
422:
421:
415:
412:
410:
407:
405:
402:
400:
397:
395:
392:
390:
387:
385:
382:
380:
377:
376:
374:
373:
369:
368:
362:
359:
357:
354:
352:
349:
347:
344:
343:
341:
340:
337:Constructions
336:
335:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
274:
271:
269:
266:
265:
263:
262:
259:
256:
255:
249:
236:
234:
221:
219:
206:
204:
191:
189:
176:
174:
161:
159:
146:
144:
131:
129:
116:
114:
101:
99:
86:
85:
83:
82:
79:
75:
71:
70:
67:
64:
63:
58:
54:
50:
46:
45:
42:
39:
38:
33:
19:
5246:
5044:Ultraproduct
4891:Model theory
4856:Independence
4792:Formal proof
4784:Proof theory
4767:
4740:
4697:real numbers
4669:second-order
4580:Substitution
4457:Metalanguage
4398:conservative
4371:Axiom schema
4315:Constructive
4285:Morse–Kelley
4251:Set theories
4230:Aleph number
4223:inaccessible
4187:
4129:Grothendieck
4013:intersection
3900:Higher-order
3888:Second-order
3834:Truth tables
3791:Venn diagram
3574:Formal proof
3468:
3464:Bourbaki, N.
3444:. Retrieved
3425:
3418:
3409:
3403:
3392:. Retrieved
3388:
3373:
3362:. Retrieved
3350:
3330:
3323:
3312:
3305:
3294:. Retrieved
3290:
3267:. Retrieved
3263:
3228:Fiber bundle
3218:Covering map
3190:
3183:
3179:
3178:surjective.
3175:
3171:
3167:
3163:
3159:
3155:
3151:
3147:
3143:
3139:
3135:
3131:
3127:
3123:
3119:
3115:
3111:
3107:
3088:
3084:
3077:
3070:
3066:
3059:
3052:
3048:
3041:
3037:
3030:
3023:
3016:
3012:
3005:
2998:
2991:
2987:
2980:
2976:
2969:
2965:
2958:
2954:
2947:
2943:
2939:
2935:
2931:
2927:
2923:
2919:
2915:
2911:
2855:
2839:
2660:
2656:
2646:Given fixed
2645:
2635:
2630:
2626:
2622:
2618:
2614:
2605:
2601:
2597:
2593:
2588:
2584:
2576:
2572:
2564:
2560:
2556:
2552:
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2516:
2508:
2503:
2499:
2495:
2488:
2478:
2474:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2429:
2425:
2420:
2414:
2410:
2402:
2397:
2391:
2387:
2381:
2377:
2373:
2367:
2363:
2359:
2353:
2349:
2345:
2338:
2330:epimorphisms
2321:
2317:
2312:
2306:
2300:
2294:
2289:
2285:
2281:
2277:
2271:
2255:
2251:
2244:
2240:
2234:
2230:
2225:
2221:
2217:
2212:
2208:
2203:
2199:
2197:
2188:
2183:
2179:
2175:
2166:
2162:
2160:
2155:
2151:
2143:
2139:
2135:
2131:
2127:
2123:
2119:
2114:
2110:
2106:
2093:
2089:
2084:
2080:
2076:
2069:
2055:
2051:
2047:
2043:
2035:
2031:
2028:right-unique
2019:
2015:
2013:
1999:
1994:is called a
1991:
1987:
1985:
1980:
1976:
1972:
1968:
1964:
1957:epimorphisms
1939:
1935:
1929:
1923:
1919:
1913:
1907:
1903:
1899:
1895:
1885:
1881:
1877:
1874:
1864:
1860:
1856:
1852:
1848:
1844:
1840:
1836:
1832:
1828:
1824:
1818:
1811:
1807:
1799:
1794:
1790:
1786:
1782:
1777:
1769:
1764:
1760:
1756:
1753:
1746:
1741:
1737:
1733:
1729:
1725:
1720:
1714:
1709:
1701:
1697:
1693:
1685:
1681:
1676:
1670:
1662:
1658:
1654:
1650:
1646:
1642:
1637:
1633:
1629:
1625:
1619:
1615:
1611:
1601:
1597:
1593:
1590:
1574:
1563:
1548:
1544:
1526:
1522:
1514:
1506:
1502:
1495:
1491:
1475:
1461:
1457:
1452:
1448:
1444:
1440:
1436:
1431:
1427:
1423:
1414:
1410:
1406:
1399:
1394:
1390:
1385:
1381:
1377:
1371:
1367:
1363:
1355:
1351:
1347:
1343:
1339:
1327:
1323:
1319:
1315:
1308:
1304:
1300:
1296:
1291:
1287:
1283:
1275:
1271:
1267:
1263:
1259:
1255:
1251:
1240:
1236:
1232:
1227:
1223:
1219:
1203:2 (that is,
1198:
1195:
1191:
1187:
1181:
1177:
1169:
1164:
1155:
1154:For any set
1139:
1135:
1131:
1127:
1126:, the point
1123:
1119:
1115:
1111:
1107:
1095:
1091:
1084:
1077:
935:
692:
690:
660:
651:
647:
641:
623:
617:
613:
611:
605:
599:
593:
583:
577:
567:
561:
557:
553:
547:
543:
539:
529:
527:one element
515:
509:
473:
469:
465:
459:
404:Higher-order
322:
56:
52:
48:
5154:Type theory
5102:undecidable
5034:Truth value
4921:equivalence
4600:non-logical
4213:Enumeration
4203:Isomorphism
4150:cardinality
4134:Von Neumann
4099:Ultrafilter
4064:Uncountable
3998:equivalence
3915:Quantifiers
3905:Fixed-point
3874:First-order
3754:Consistency
3739:Proposition
3716:Traditional
3687:Lindström's
3677:Compactness
3619:Type theory
3564:Cardinality
3223:Enumeration
2666:cardinality
2559:(~) :
2274:composition
2118:satisfying
2072:cardinality
2060:right-total
1945:composition
1911:, whenever
1875:A function
1863:"reverses"
1667:composition
1421:defined by
1375:defined by
1313:real number
1295:defined by
1249:real number
1231:defined by
1186:defined by
675:composition
663:restricting
462:mathematics
389:Multivalued
351:Composition
346:Restriction
5269:Categories
4965:elementary
4658:arithmetic
4526:Quantifier
4504:functional
4376:Expression
4094:Transitive
4038:identities
4023:complement
3956:hereditary
3939:Set theory
3500:2004110815
3446:2009-11-25
3394:2019-12-06
3364:2013-05-11
3296:2019-12-07
3269:2019-12-07
3244:References
3196:are shown.
3040:such that
2990:There are
2968:such that
2930:such that
2856:surjective
2840:surjective
2814:denotes a
2583:, and let
2567:/~ be the
2511:/~ be the
2385:such that
2024:left-total
1839:such that
1827:such that
1641:for every
1560:Properties
1537:projection
1513:of degree
1455:such that
1397:such that
1258:such that
681:Definition
614:surjective
537:such that
470:surjection
323:Surjective
313:Measurable
308:Continuous
283:Polynomial
18:Surjective
5236:Supertask
5139:Recursion
5097:decidable
4931:saturated
4909:of models
4832:deductive
4827:axiomatic
4747:Hilbert's
4734:Euclidean
4715:canonical
4638:axiomatic
4570:Signature
4499:Predicate
4388:Extension
4310:Ackermann
4235:Operation
4114:Universal
4104:Recursive
4079:Singleton
4074:Inhabited
4059:Countable
4049:Types of
4033:power set
4003:partition
3920:Predicate
3866:Predicate
3781:Syllogism
3771:Soundness
3744:Inference
3734:Tautology
3636:paradoxes
3466:(2004) .
3233:Index set
2946:There is
2469:to which
2440:partition
2407:preimages
2341:injection
2193:injective
1949:morphisms
1570:injective
1566:bijective
1472:function
1336:bijective
1332:injective
1245:bijective
1026:∈
1020:∃
1010:∈
1004:∀
957:→
951::
918:↠
912::
901:), as in
625:bijective
619:injective
612:The term
328:Bijective
318:Injective
293:Algebraic
72:Types by
5221:Logicism
5214:timeline
5190:Concrete
5049:Validity
5019:T-schema
5012:Kripke's
5007:Tarski's
5002:semantic
4992:Strength
4941:submodel
4936:spectrum
4904:function
4752:Tarski's
4741:Elements
4728:geometry
4684:Robinson
4605:variable
4590:function
4563:spectrum
4553:Sentence
4509:variable
4452:Language
4405:Relation
4366:Automata
4356:Alphabet
4340:language
4194:-jection
4172:codomain
4158:Function
4119:Universe
4089:Infinite
3993:Relation
3776:Validity
3766:Argument
3664:theorem,
3331:Bourbaki
3316:, Tripod
3202:See also
3122:, where
3114: :
2751:, where
2621:). Then
2592: :
2498: :
2491:quotient
2436:disjoint
2376: :
2362: :
2348: :
2334:category
2178: :
2142:exists.
2134:for all
2109: :
2079: :
1971:meaning
1953:category
1902: :
1880: :
1804:preimage
1798:. Thus,
1759: :
1614: :
1596: :
1499:matrices
1409: :
1366: :
1354:, −2 ≤
1342:= 2 is {
1286: :
1222: :
1208:integers
1184:→ {0, 1}
1180: :
1089:codomain
1069:Examples
896:↠
705:codomain
697:function
556: :
525:at least
521:codomain
506:function
409:Morphism
394:Implicit
298:Analytic
288:Rational
273:Identity
268:Constant
78:codomain
55: (
41:Function
5163:Related
4960:Diagram
4858: (
4837:Hilbert
4822:Systems
4817:Theorem
4695:of the
4640:systems
4420:Formula
4415:Grammar
4331: (
4275:General
3988:Forcing
3973:Element
3893:Monadic
3668:paradox
3609:Theorem
3545:General
3076:), and
2822:Gallery
2664:. The
2445:. Then
2332:in any
2328:to any
2292:, then
1996:section
1933:, then
1821:gallery
1780:, then
1738:reverse
1706:inverse
1665:if the
1581:mapping
1539:from a
1521:of all
1278:− 1)/2.
1098:is the
972:, then
571:is the
504:) is a
414:Functor
384:Partial
361:Inverse
4926:finite
4689:Skolem
4642:
4617:Theory
4585:Symbol
4575:String
4558:atomic
4435:ground
4430:closed
4425:atomic
4381:ground
4344:syntax
4240:binary
4167:domain
4084:Finite
3849:finite
3707:Logics
3666:
3614:Theory
3498:
3488:
3437:
3338:
3180:Right:
3029:, and
2988:Right:
2555:. Let
2423:is in
2419:where
2171:finite
1843:(4) =
1774:subset
1346:= −1,
1158:, the
1082:domain
893:
729:domain
699:whose
646:means
634:French
589:unique
535:domain
303:Smooth
278:Linear
74:domain
4916:Model
4664:Peano
4521:Proof
4361:Arity
4290:Naive
4177:image
4109:Fuzzy
4069:Empty
4018:union
3963:Class
3604:Model
3594:Lemma
3552:Axiom
3385:(PDF)
3359:(PDF)
3172:Left:
2944:Left:
2638:(~).
2596:/~ →
2254:| = |
2224:onto
2154:| ≤ |
2130:)) =
1977:above
1951:of a
1793:)) =
1772:is a
1636:)) =
1577:graph
1519:group
1239:) = 2
1106:) of
1104:range
1100:image
1080:from
850:with
727:with
701:image
695:is a
656:image
652:above
573:image
472:, or
399:Space
5039:Type
4842:list
4646:list
4623:list
4612:Term
4546:rank
4440:open
4334:list
4146:Maps
4051:sets
3910:Free
3880:list
3630:list
3557:list
3496:LCCN
3486:ISBN
3435:ISBN
3336:ISBN
3189:and
3004:and
2650:and
2613:) =
2539:) =
2438:and
2280:and
2272:The
2238:and
2202:and
2169:are
2165:and
2150:of |
2070:The
2054:and
2034:and
2026:and
1973:over
1684:and
1535:The
1486:The
1468:The
1430:) =
1402:= −1
1384:) =
1303:) =
1274:is (
1266:) =
1205:even
1194:) =
891:21A0
648:over
622:and
546:) =
464:, a
76:and
4726:of
4708:of
4656:of
4188:Sur
4162:Map
3969:Ur-
3951:Set
3478:doi
3158:in
3150:in
3134:),
3058:),
3036:in
3011:in
2986:).
2964:in
2953:in
2926:in
2918:in
2575:in
2515:of
2191:is
2138:in
1998:of
1969:ἐπί
1965:epi
1776:of
1754:If
1728:of
1708:of
1696:of
1680:of
1645:in
1623:if
1391:not
1389:is
1322:− 3
1307:− 3
1212:odd
1200:mod
1168:on
1122:in
1112:not
1087:to
940:If
830:in
790:in
650:or
643:sur
587:be
460:In
5271::
5112:NP
4736::
4730::
4660::
4337:),
4192:Bi
4184:In
3494:.
3484:.
3472:.
3433:.
3387:.
3289:.
3278:^
3262:.
3251:^
3170:.
3126:=
3118:→
3094:).
3083:=
3065:=
3047:=
3022:,
2997:,
2975:=
2934:=
2818:.
2659:↠
2634:o
2625:=
2563:→
2527:~
2523::
2502:→
2390:=
2380:→
2366:→
2352:→
2336:.
2264:.
2258:|,
2243:≤
2233:≤
2211:≤
2195:.
2182:→
2113:→
2083:→
2062:.
2006:.
1983:.
1981:on
1979:,
1975:,
1938:=
1922:=
1906:→
1884:→
1867:.
1816:.
1763:→
1751:.
1740:"
1618:→
1600:→
1572:.
1547:×
1460:=
1437:is
1418:≥0
1413:→
1370:→
1326:−
1290:→
1226:→
1162:id
933:.
889:U+
691:A
609:.
560:→
497:uː
242:→
227:→
214:→
197:→
182:→
169:→
152:→
139:→
122:→
120:𝔹
107:→
105:𝔹
96:𝔹
94:→
51:↦
5192:/
5107:P
4862:)
4648:)
4644:(
4541:∀
4536:!
4531:∃
4492:=
4487:↔
4482:→
4477:∧
4472:∨
4467:¬
4190:/
4186:/
4160:/
3971:)
3967:(
3854:∞
3844:3
3632:)
3530:e
3523:t
3516:v
3502:.
3480::
3449:.
3397:.
3367:.
3344:.
3318:.
3299:.
3272:.
3194:2
3191:X
3187:1
3184:X
3176:f
3168:y
3164:x
3160:X
3156:x
3152:Y
3148:y
3144:f
3140:Y
3136:X
3132:x
3130:(
3128:f
3124:y
3120:Y
3116:X
3112:f
3092:3
3089:x
3087:(
3085:f
3081:3
3078:y
3074:2
3071:x
3069:(
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3060:y
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3024:x
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3009:3
3006:y
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2999:y
2995:1
2992:y
2984:0
2981:x
2979:(
2977:f
2973:0
2970:y
2966:X
2962:0
2959:x
2955:Y
2951:0
2948:y
2940:x
2938:(
2936:f
2932:y
2928:X
2924:x
2920:Y
2916:y
2800:}
2793:|
2789:B
2785:|
2776:|
2772:A
2768:|
2761:{
2737:}
2730:|
2726:B
2722:|
2713:|
2709:A
2705:|
2698:{
2693:!
2689:|
2685:B
2681:|
2661:B
2657:A
2652:B
2648:A
2636:P
2631:P
2627:f
2623:f
2619:x
2617:(
2615:f
2611:~
2609:(
2606:P
2602:f
2598:B
2594:A
2589:P
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2581:~
2577:A
2573:x
2565:A
2561:A
2557:P
2553:f
2549:A
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2543:(
2541:f
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2517:A
2509:A
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2500:A
2496:f
2479:g
2475:f
2471:h
2467:Z
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2430:X
2428:(
2426:h
2421:z
2417:)
2415:z
2413:(
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2364:X
2360:f
2354:Z
2350:X
2346:h
2322:g
2318:f
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2307:f
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2278:f
2256:X
2252:Y
2250:|
2245:X
2241:Y
2235:Y
2231:X
2226:X
2222:Y
2218:X
2213:Y
2209:X
2204:Y
2200:X
2189:f
2184:Y
2180:X
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2167:Y
2163:X
2156:X
2152:Y
2144:g
2140:Y
2136:y
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2128:y
2126:(
2124:g
2122:(
2120:f
2115:X
2111:Y
2107:g
2094:Y
2090:X
2085:Y
2081:X
2077:f
2056:Y
2052:X
2048:Y
2044:X
2036:Y
2032:X
2020:Y
2016:X
2000:f
1992:f
1988:g
1940:h
1936:g
1930:f
1927:o
1924:h
1920:f
1917:o
1914:g
1908:Z
1904:Y
1900:h
1898:,
1896:g
1886:Y
1882:X
1878:f
1865:g
1861:f
1857:C
1855:(
1853:g
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1845:C
1841:f
1837:f
1833:C
1831:(
1829:g
1825:g
1814:)
1812:B
1810:(
1808:f
1800:B
1795:B
1791:B
1789:(
1787:f
1785:(
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1778:Y
1770:B
1765:Y
1761:X
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1742:g
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1715:g
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1698:g
1694:Y
1686:f
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1677:g
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1671:f
1663:f
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1649:(
1647:Y
1643:y
1638:y
1634:y
1632:(
1630:g
1628:(
1626:f
1620:Y
1616:X
1612:f
1602:X
1598:Y
1594:g
1549:B
1545:A
1527:n
1525:×
1523:n
1515:n
1507:n
1505:×
1503:n
1496:n
1494:×
1492:n
1476:R
1465:.
1462:y
1458:x
1453:X
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1441:y
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1428:x
1426:(
1424:g
1415:R
1411:R
1407:g
1400:x
1395:x
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1380:(
1378:g
1372:R
1368:R
1364:g
1356:y
1352:y
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1344:x
1340:y
1328:y
1324:x
1320:x
1316:y
1309:x
1305:x
1301:x
1299:(
1297:f
1292:R
1288:R
1284:f
1276:y
1272:x
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1264:x
1262:(
1260:f
1256:x
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1237:x
1235:(
1233:f
1228:R
1224:R
1220:f
1196:n
1192:n
1190:(
1188:f
1182:Z
1178:f
1170:X
1165:X
1156:X
1150:.
1140:f
1136:Y
1132:x
1130:(
1128:f
1124:X
1120:x
1116:Y
1108:f
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1052:y
1049:=
1046:)
1043:x
1040:(
1037:f
1032:,
1029:X
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1016:,
1013:Y
1007:y
980:f
960:Y
954:X
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915:X
909:f
873:y
870:=
867:)
864:x
861:(
858:f
838:X
818:x
798:Y
778:y
758:Y
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606:Y
600:X
594:f
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578:X
568:Y
562:Y
558:X
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542:(
540:f
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500:/
494:t
491:.
488:n
485:ɒ
482:ˈ
479:/
449:e
442:t
435:v
356:λ
245:X
240:ℂ
230:X
225:ℂ
216:ℂ
211:X
200:X
195:ℝ
185:X
180:ℝ
171:ℝ
166:X
155:X
150:ℤ
141:ℤ
136:X
125:X
110:X
91:X
59:)
57:x
53:f
49:x
34:.
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