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can be applied equally to maps and functions, with the same meaning. All these usages can be applied to "maps" as general functions or as functions with special properties.
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292:" or "arrow", which is a structure-respecting function and thus may imply more structure than "function" does. For example, a morphism
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329:(i.e. a morphism that can be viewed as a function) carries with it the information of its domain (the source
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144:, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "
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A map is a function, as in the association of any of the four colored shapes in X to its color in Y
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with a more restricted meaning, namely as a map that is restricted to apply to numbers only.
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often refers to a function from a set to itself. There are also a few less common uses in
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in its general sense. These terms may have originated as from the process of making a
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796:"Functions or Mapping | Learning Mapping | Function as a Special Kind of Relation"
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575: â Function that maps a function and its arguments to the function value
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2229:
2109:
2057:
1966:
1902:
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1816:
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628: â Mapping which preserves all topological properties of a given space
219:
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119:
85:
77:
37:
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645: â Group of isotopy classes of a topological automorphism group
190:
Maps of certain kinds have been given specific names. These include
76:
may be used to distinguish some special types of functions, such as
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1912:
1617:
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203:
168:
149:
101:
2262:
1054:
195:
1806:
1152:
997:
572:
115:
651: â Group whose operation is composition of permutations
288:
In category theory, "map" is often used as a synonym for "
62:
514:. In this sense, the function does not capture the set
587:
540:
520:
494:
453:
427:
407:
375:
355:
335:
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167:, use "function" only to refer to maps in which the
605:
555:
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439:
413:
393:
361:
341:
317:
2712:
369:). In the widely used definition of a function
349:of the morphism) and its codomain (the target
968:
534:that is used as the codomain; only the range
872:(2nd ed.). Addison-Wesley. p. 83.
622: â Properties of mathematical functions
1160:
975:
961:
845:"Mapping, Mathematical | Encyclopedia.com"
136:In many branches of mathematics, the term
899:. Cambridge University Press. p. 2.
305:
694:, p. 30. Some authors use the term
36:
892:
738:
657: â Morphism of algebraic varieties
92:may have this meaning or it may mean a
69:the Earth surface to a sheet of paper.
14:
2713:
982:
923:
766:
691:
171:is a set of numbers (i.e. a subset of
956:
818:
867:
710:
708:
706:
704:
125:
620:Bijection, injection and surjection
27:Function, homomorphism, or morphism
24:
896:An Introduction to Category Theory
767:Stacho, Juraj (October 31, 2007).
25:
2737:
948:
812:
701:
108:can be used interchangeably, but
2694:
655:Regular map (algebraic geometry)
639:â commonly pronounced "maps to"
563:is determined by the function.
277:
917:
886:
861:
837:
788:
760:
746:. Addison-Wesley. p. 35.
732:
668:
591:
550:
544:
475:
472:
466:
454:
385:
309:
250:. Related terminology such as
13:
1:
2655:History of mathematical logic
690:are often used synonymously.
661:
2726:Basic concepts in set theory
2580:Primitive recursive function
769:"Function, one-to-one, onto"
606:{\displaystyle x\mapsto x+1}
447:consisting of all the pairs
187:for more general functions.
7:
566:
10:
2742:
1644:SchröderâBernstein theorem
1371:Monadic predicate calculus
1030:Foundations of mathematics
318:{\displaystyle f:\,X\to Y}
281:
235:discrete dynamical systems
129:
29:
2690:
2677:Philosophy of mathematics
2626:Automated theorem proving
2608:
2503:
2335:
2228:
2080:
1797:
1773:
1751:Von NeumannâBernaysâGödel
1696:
1590:
1494:
1392:
1383:
1310:
1245:
1151:
1073:
990:
440:{\displaystyle X\times Y}
481:{\displaystyle (x,f(x))}
394:{\displaystyle f:X\to Y}
183:), and reserve the term
2327:Self-verifying theories
2148:Tarski's axiomatization
1099:Tarski's undefinability
1094:incompleteness theorems
720:Encyclopedia Britannica
716:"Mapping | mathematics"
100:, a map may refer to a
2721:Functions and mappings
2701:Mathematics portal
2312:Proof of impossibility
1960:propositional variable
1270:Propositional calculus
607:
557:
528:
508:
507:{\displaystyle x\in X}
482:
441:
415:
395:
363:
343:
319:
163:Some authors, such as
132:Function (mathematics)
42:
2570:Kolmogorov complexity
2523:Computably enumerable
2423:Model complete theory
2215:Principia Mathematica
1275:Propositional formula
1104:BanachâTarski paradox
825:mathworld.wolfram.com
744:Mathematical Analysis
608:
558:
529:
509:
483:
442:
416:
396:
364:
344:
320:
154:linear transformation
84:is a homomorphism of
40:
18:Mapping (mathematics)
2518:ChurchâTuring thesis
2505:Computability theory
1714:continuum hypothesis
1232:Square of opposition
1090:Gödel's completeness
893:Simmons, H. (2011).
868:Lang, Serge (1971).
849:www.encyclopedia.com
632:List of chaotic maps
585:
556:{\displaystyle f(X)}
538:
518:
492:
451:
425:
405:
373:
353:
333:
296:
32:map (disambiguation)
30:For other uses, see
2672:Mathematical object
2563:P versus NP problem
2528:Computable function
2322:Reverse mathematics
2248:Logical consequence
2125:primitive recursive
2120:elementary function
1893:Free/bound variable
1746:TarskiâGrothendieck
1265:Logical connectives
1195:Logical equivalence
1045:Logical consequence
933:. Springer-Verlag.
819:Weisstein, Eric W.
643:Mapping class group
229:, a map denotes an
146:continuous function
2470:Transfer principle
2433:Semantics of logic
2418:Categorical theory
2394:Non-standard model
1908:Logical connective
1035:Information theory
984:Mathematical logic
603:
553:
524:
504:
478:
437:
411:
391:
359:
339:
315:
231:evolution function
140:is used to mean a
43:
2708:
2707:
2640:Abstract category
2443:Theories of truth
2253:Rule of inference
2243:Natural deduction
2224:
2223:
1769:
1768:
1474:Cartesian product
1379:
1378:
1285:Many-valued logic
1260:Boolean functions
1143:Russell's paradox
1118:diagonal argument
1015:First-order logic
940:978-0-387-90092-6
906:978-1-139-50332-7
649:Permutation group
527:{\displaystyle Y}
414:{\displaystyle f}
362:{\displaystyle Y}
342:{\displaystyle X}
327:concrete category
227:dynamical systems
225:In the theory of
126:Maps as functions
94:linear polynomial
88:, while the term
80:. For example, a
16:(Redirected from
2733:
2699:
2698:
2650:History of logic
2645:Category of sets
2538:Decision problem
2317:Ordinal analysis
2258:Sequent calculus
2156:Boolean algebras
2096:
2095:
2070:
2041:logical/constant
1795:
1794:
1781:
1704:ZermeloâFraenkel
1455:Set operations:
1390:
1389:
1327:
1158:
1157:
1138:LöwenheimâSkolem
1025:Formal semantics
977:
970:
963:
954:
953:
944:
930:Naive Set Theory
911:
910:
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883:
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637:Maplet arrow (âŠ)
613:, also known as
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247:partial function
63:geographical map
21:
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2710:
2709:
2704:
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2686:
2631:Category theory
2621:Algebraic logic
2604:
2575:Lambda calculus
2513:Church encoding
2499:
2475:Truth predicate
2331:
2297:Complete theory
2220:
2089:
2085:
2081:
2076:
2068:
1788: and
1784:
1779:
1765:
1741:New Foundations
1709:axiom of choice
1692:
1654:Gödel numbering
1594: and
1586:
1490:
1375:
1325:
1306:
1255:Boolean algebra
1241:
1205:Equiconsistency
1170:Classical logic
1147:
1128:Halting problem
1116: and
1092: and
1080: and
1079:
1074:Theorems (
1069:
986:
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951:
941:
925:Halmos, Paul R.
920:
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421:is a subset of
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233:used to create
216:representations
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98:category theory
90:linear function
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2595:Turing machine
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2533:Computable set
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2428:Satisfiability
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2345:Interpretation
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2166:minimal axioms
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1775:Formal systems
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1664:Large cardinal
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1429:Extensionality
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1424:Ordinal number
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1108:Cantor's
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949:External links
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912:
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878:
870:Linear Algebra
860:
836:
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800:Math Only Math
787:
776:cs.toronto.edu
759:
752:
740:Apostol, T. M.
731:
700:
684:correspondence
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579:Arrow notation
576:
573:Apply function
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282:Main article:
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158:linear algebra
130:Main article:
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111:transformation
106:transformation
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2622:
2619:
2617:
2614:
2613:
2611:
2607:
2601:
2598:
2596:
2593:
2591:
2590:Recursive set
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2540:
2539:
2536:
2534:
2531:
2529:
2526:
2524:
2521:
2519:
2516:
2514:
2511:
2510:
2508:
2506:
2502:
2496:
2493:
2491:
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2463:
2459:
2456:
2454:
2451:
2449:
2446:
2445:
2444:
2441:
2439:
2436:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2412:
2409:
2408:
2407:
2404:
2400:
2399:of arithmetic
2397:
2396:
2395:
2392:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
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2356:
2353:
2351:
2348:
2347:
2346:
2343:
2342:
2340:
2338:
2334:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2307:
2306:from ZFC
2303:
2300:
2298:
2295:
2289:
2286:
2285:
2284:
2281:
2279:
2276:
2274:
2271:
2270:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2235:
2233:
2231:
2227:
2217:
2216:
2212:
2211:
2206:
2205:non-Euclidean
2203:
2199:
2196:
2194:
2191:
2189:
2188:
2184:
2183:
2181:
2178:
2177:
2175:
2171:
2167:
2164:
2162:
2159:
2158:
2157:
2153:
2149:
2146:
2145:
2144:
2140:
2136:
2133:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2107:
2105:
2101:
2100:
2098:
2093:
2087:
2082:Example
2079:
2071:
2066:
2065:
2064:
2061:
2059:
2056:
2052:
2049:
2047:
2044:
2042:
2039:
2037:
2034:
2033:
2032:
2029:
2027:
2024:
2022:
2019:
2017:
2014:
2010:
2007:
2005:
2002:
2001:
2000:
1997:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1974:
1973:
1970:
1968:
1965:
1961:
1958:
1956:
1953:
1951:
1948:
1947:
1946:
1943:
1939:
1936:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1916:
1914:
1911:
1910:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1887:
1884:
1882:
1879:
1877:
1874:
1872:
1869:
1868:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1845:
1842:
1840:
1839:by definition
1837:
1836:
1835:
1832:
1828:
1825:
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1823:
1820:
1818:
1815:
1813:
1810:
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1787:
1782:
1776:
1772:
1762:
1759:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1726:KripkeâPlatek
1724:
1722:
1719:
1715:
1712:
1710:
1707:
1706:
1705:
1702:
1701:
1699:
1695:
1687:
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1683:
1682:
1679:
1677:
1674:
1670:
1667:
1666:
1665:
1662:
1660:
1657:
1655:
1652:
1650:
1647:
1645:
1642:
1639:
1635:
1631:
1628:
1624:
1621:
1619:
1616:
1614:
1611:
1610:
1609:
1605:
1602:
1601:
1599:
1597:
1593:
1589:
1581:
1578:
1576:
1573:
1571:
1570:constructible
1568:
1567:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
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1523:
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1460:
1457:
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1372:
1369:
1367:
1364:
1362:
1359:
1357:
1354:
1352:
1349:
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1344:
1340:
1337:
1336:
1335:
1332:
1328:
1323:
1322:
1321:
1318:
1317:
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1309:
1301:
1298:
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1281:
1278:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1253:
1252:
1250:
1248:
1247:Propositional
1244:
1238:
1235:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1206:
1203:
1202:
1201:
1198:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1176:
1175:Logical truth
1173:
1171:
1168:
1167:
1165:
1163:
1159:
1156:
1154:
1150:
1144:
1141:
1139:
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1129:
1126:
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1121:
1119:
1115:
1111:
1107:
1105:
1102:
1100:
1097:
1095:
1091:
1088:
1087:
1085:
1083:
1077:
1072:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1004:
1001:
1000:
999:
996:
995:
993:
989:
985:
978:
973:
971:
966:
964:
959:
958:
955:
942:
936:
932:
931:
926:
922:
921:
908:
902:
898:
897:
889:
881:
879:0-201-04211-8
875:
871:
864:
850:
846:
840:
826:
822:
815:
801:
797:
791:
777:
770:
763:
755:
753:0-201-00288-4
749:
745:
741:
735:
721:
717:
711:
709:
707:
705:
697:
693:
689:
685:
681:
677:
671:
667:
656:
653:
650:
647:
644:
641:
638:
635:
633:
630:
627:
626:Homeomorphism
624:
621:
618:
616:
600:
597:
594:
588:
580:
577:
574:
571:
570:
564:
547:
541:
521:
501:
498:
495:
469:
463:
460:
457:
434:
431:
428:
408:
388:
382:
379:
376:
356:
336:
328:
312:
306:
302:
299:
291:
285:
275:
273:
272:
267:
266:
261:
260:
255:
254:
249:
248:
243:
238:
236:
232:
228:
223:
221:
217:
213:
209:
205:
201:
197:
193:
192:homomorphisms
188:
186:
182:
181:
176:
175:
170:
166:
161:
159:
155:
151:
147:
143:
139:
133:
123:
121:
117:
113:
112:
107:
103:
99:
95:
91:
87:
86:vector spaces
83:
79:
78:homomorphisms
75:
70:
68:
64:
60:
56:
52:
48:
39:
33:
19:
2692:
2490:Ultraproduct
2337:Model theory
2302:Independence
2238:Formal proof
2230:Proof theory
2213:
2186:
2143:real numbers
2115:second-order
2026:Substitution
1903:Metalanguage
1844:conservative
1817:Axiom schema
1761:Constructive
1731:MorseâKelley
1697:Set theories
1676:Aleph number
1669:inaccessible
1607:
1591:
1575:Grothendieck
1459:intersection
1346:Higher-order
1334:Second-order
1280:Truth tables
1237:Venn diagram
1020:Formal proof
929:
895:
888:
869:
863:
852:. Retrieved
848:
839:
828:. Retrieved
824:
814:
803:. Retrieved
799:
790:
779:. Retrieved
775:
762:
743:
734:
723:. Retrieved
719:
695:
687:
683:
679:
675:
670:
614:
287:
278:As morphisms
269:
263:
257:
251:
245:
241:
239:
224:
220:group theory
189:
184:
179:
173:
162:
137:
135:
120:graph theory
109:
105:
73:
71:
66:
54:
50:
44:
2600:Type theory
2548:undecidable
2480:Truth value
2367:equivalence
2046:non-logical
1659:Enumeration
1649:Isomorphism
1596:cardinality
1580:Von Neumann
1545:Ultrafilter
1510:Uncountable
1444:equivalence
1361:Quantifiers
1351:Fixed-point
1320:First-order
1200:Consistency
1185:Proposition
1162:Traditional
1133:Lindström's
1123:Compactness
1065:Type theory
1010:Cardinality
918:Works cited
692:Halmos 1970
242:partial map
104:. The term
47:mathematics
2715:Categories
2411:elementary
2104:arithmetic
1972:Quantifier
1950:functional
1822:Expression
1540:Transitive
1484:identities
1469:complement
1402:hereditary
1385:Set theory
854:2019-12-06
830:2019-12-06
805:2019-12-06
781:2019-12-06
725:2019-12-06
674:The words
662:References
271:continuous
200:isometries
165:Serge Lang
82:linear map
2682:Supertask
2585:Recursion
2543:decidable
2377:saturated
2355:of models
2278:deductive
2273:axiomatic
2193:Hilbert's
2180:Euclidean
2161:canonical
2084:axiomatic
2016:Signature
1945:Predicate
1834:Extension
1756:Ackermann
1681:Operation
1560:Universal
1550:Recursive
1525:Singleton
1520:Inhabited
1505:Countable
1495:Types of
1479:power set
1449:partition
1366:Predicate
1312:Predicate
1227:Syllogism
1217:Soundness
1190:Inference
1180:Tautology
1082:paradoxes
592:↦
499:∈
432:×
386:→
310:→
265:injective
208:operators
72:The term
2667:Logicism
2660:timeline
2636:Concrete
2495:Validity
2465:T-schema
2458:Kripke's
2453:Tarski's
2448:semantic
2438:Strength
2387:submodel
2382:spectrum
2350:function
2198:Tarski's
2187:Elements
2174:geometry
2130:Robinson
2051:variable
2036:function
2009:spectrum
1999:Sentence
1955:variable
1898:Language
1851:Relation
1812:Automata
1802:Alphabet
1786:language
1640:-jection
1618:codomain
1604:Function
1565:Universe
1535:Infinite
1439:Relation
1222:Validity
1212:Argument
1110:theorem,
927:(1970).
742:(1981).
696:function
688:operator
581:â e.g.,
567:See also
290:morphism
284:Morphism
259:codomain
212:analysis
204:geometry
169:codomain
150:topology
142:function
102:morphism
59:function
2609:Related
2406:Diagram
2304: (
2283:Hilbert
2268:Systems
2263:Theorem
2141:of the
2086:systems
1866:Formula
1861:Grammar
1777: (
1721:General
1434:Forcing
1419:Element
1339:Monadic
1114:paradox
1055:Theorem
991:General
680:mapping
196:algebra
185:mapping
160:, etc.
67:mapping
55:mapping
2372:finite
2135:Skolem
2088:
2063:Theory
2031:Symbol
2021:String
2004:atomic
1881:ground
1876:closed
1871:atomic
1827:ground
1790:syntax
1686:binary
1613:domain
1530:Finite
1295:finite
1153:Logics
1112:
1060:Theory
937:
903:
876:
750:
686:, and
268:, and
253:domain
2362:Model
2110:Peano
1967:Proof
1807:Arity
1736:Naive
1623:image
1555:Fuzzy
1515:Empty
1464:union
1409:Class
1050:Model
1040:Lemma
998:Axiom
821:"Map"
772:(PDF)
325:in a
244:is a
156:" in
152:, a "
148:" in
116:logic
96:. In
57:is a
2485:Type
2288:list
2092:list
2069:list
2058:Term
1992:rank
1886:open
1780:list
1592:Maps
1497:sets
1356:Free
1326:list
1076:list
1003:list
935:ISBN
901:ISBN
874:ISBN
748:ISBN
488:for
222:.
214:and
118:and
49:, a
2172:of
2154:of
2102:of
1634:Sur
1608:Map
1415:Ur-
1397:Set
676:map
615:map
237:.
218:in
210:in
202:in
194:in
177:or
138:map
74:map
53:or
51:map
45:In
2717::
2558:NP
2182::
2176::
2106::
1783:),
1638:Bi
1630:In
847:.
823:.
798:.
774:.
718:.
703:^
682:,
678:,
401:,
262:,
256:,
240:A
206:,
198:,
122:.
65::
2638:/
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2094:)
2090:(
1987:â
1982:!
1977:â
1938:=
1933:â
1928:â
1923:â§
1918:âš
1913:ÂŹ
1636:/
1632:/
1606:/
1417:)
1413:(
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976:e
969:t
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857:.
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784:.
756:.
728:.
601:1
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548:X
545:(
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470:x
467:(
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455:(
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383:X
380::
377:f
357:Y
337:X
313:Y
307:X
303::
300:f
180:C
174:R
34:.
20:)
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