Codomain From Wikipedia, the free encyclopedia Contents 1 Algebra of sets 1 1.1 Fundamentals ............................................. 1 1.2 The fundamental laws of set algebra .................................. 1 1.3 The principle of duality ........................................ 2 1.4 Some additional laws for unions and intersections .......................... 2 1.5 Some additional laws for complements ................................ 3 1.6 The algebra of inclusion ........................................ 3 1.7 The algebra of relative complements ................................. 4 1.8 See also ................................................ 5 1.9 References ............................................... 5 1.10 External links ............................................. 5 2 Axiom of choice 6 2.1 Statement ............................................... 6 2.1.1 Nomenclature ZF, AC, and ZFC ............................... 7 2.1.2 Variants ............................................ 7 2.1.3 Restriction to finite sets .................................... 7 2.2 Usage ................................................. 8 2.3 Examples ............................................... 8 2.4 Criticism and acceptance ....................................... 8 2.5 In constructive mathematics ..................................... 9 2.6 Independence ............................................. 10 2.7 Stronger axioms ............................................ 10 2.8 Equivalents .............................................. 10 2.8.1 Category theory ....................................... 11 2.9 Weaker forms ............................................. 12 2.9.1 Results requiring AC (or weaker forms) but weaker than it .................. 12 2.10 Stronger forms of the negation of AC ................................. 13 2.11 Statements consistent with the negation of AC ............................ 13 2.12 Quotes ................................................. 14 2.13 Notes ................................................. 14 2.14 References ............................................... 16 2.15 External links ............................................. 16 i ii CONTENTS 3 Bijection 18 3.1 Definition ............................................... 19 3.2 Examples ............................................... 19 3.2.1 Batting line-up of a baseball team ............................... 19 3.2.2 Seats and students of a classroom ............................... 19 3.3 More mathematical examples and some non-examples ........................ 20 3.4 Inverses ................................................ 20 3.5 Composition .............................................. 20 3.6 Bijections and cardinality ....................................... 20 3.7 Properties ............................................... 21 3.8 Bijections and category theory ..................................... 21 3.9 Generalization to partial functions ................................... 22 3.10 Contrast with ............................................. 22 3.11 See also ................................................ 22 3.12 Notes ................................................. 22 3.13 References ............................................... 23 3.14 External links ............................................. 23 4 Binary relation 24 4.1 Formal definition ........................................... 24 4.1.1 Is a relation more than its graph? ............................... 25 4.1.2 Example ............................................ 25 4.2 Special types of binary relations .................................... 25 4.2.1 Difunctional ......................................... 27 4.3 Relations over a set .......................................... 27 4.4 Operations on binary relations ..................................... 28 4.4.1 Complement ......................................... 29 4.4.2 Restriction .......................................... 29 4.4.3 Algebras, categories, and rewriting systems ......................... 30 4.5 Sets versus classes ........................................... 30 4.6 The number of binary relations .................................... 30 4.7 Examples of common binary relations ................................. 31 4.8 See also ................................................ 31 4.9 Notes ................................................. 31 4.10 References ............................................... 32 4.11 External links ............................................. 33 5 Cardinal number 34 5.1 History ................................................. 34 5.2 Motivation .............................................. 36 5.3 Formal definition ........................................... 37 5.4 Cardinal arithmetic .......................................... 38 CONTENTS iii 5.4.1 Successor cardinal ...................................... 38 5.4.2 Cardinal addition ....................................... 38 5.4.3 Cardinal multiplication .................................... 39 5.4.4 Cardinal exponentiation ................................... 39 5.5 The continuum hypothesis ....................................... 40 5.6 See also ................................................ 40 5.7 Notes ................................................. 40 5.8 References ............................................... 40 5.9 External links ............................................. 41 6 Cardinality 42 6.1 Comparing sets ............................................ 42 6.1.1 Definition 1: | A | = | B | ................................... 42 6.1.2 Definition 2: | A | ≤ | B | ................................... 42 6.1.3 Definition 3: | A | < | B | ................................... 42 6.2 Cardinal numbers ........................................... 43 6.3 Finite, countable and uncountable sets ................................ 44 6.4 Infinite sets .............................................. 44 6.4.1 Cardinality of the continuum ................................. 44 6.5 Examples and properties ........................................ 45 6.6 Union and intersection ......................................... 45 6.7 See also ................................................ 46 6.8 References ............................................... 46 7 Cartesian product 47 7.1 Examples ............................................... 48 7.1.1 A deck of cards ....................................... 48 7.1.2 A two-dimensional coordinate system ............................ 48 7.2 Most common implementation (set theory) .............................. 48 7.2.1 Non-commutativity and non-associativity .......................... 49 7.2.2 Intersections, unions, and subsets ............................... 50 7.2.3 Cardinality .......................................... 51 7.3 n-ary product ............................................. 51 7.3.1 Cartesian power ....................................... 51 7.3.2 Finite n-ary product ..................................... 51 7.3.3 Infinite products ....................................... 52 7.4 Other forms .............................................. 52 7.4.1 Abbreviated form ....................................... 52 7.4.2 Cartesian product of functions ................................ 52 7.5 Definitions outside of Set theory ................................... 53 7.5.1 Category theory ........................................ 53 7.5.2 Graph theory ......................................... 53 iv CONTENTS 7.6 See also ................................................ 53 7.7 References .............................................. 53 7.8 External links ............................................. 54 8 Class (set theory) 55 8.1 Examples ............................................... 55 8.2 Paradoxes ............................................... 55 8.3 Classes in formal set theories ..................................... 55 8.4 References ............................................... 56 8.5 External links ............................................. 56 9 Codomain 57 9.1 Examples ............................................... 58 9.2 See also ................................................ 59 9.3 Notes ................................................. 59 9.4 References ............................................... 59 10 Complement (set theory) 60 10.1 Relative complement ......................................... 60 10.2 Absolute complement ......................................... 61 10.3 Notation ................................................ 62 10.4 Complements in various programming languages ........................... 62 10.5 See also ................................................ 64 10.6 References .............................................. 64 10.7 External links ............................................. 65 11 Disjoint sets 66 11.1 Generalizations ............................................ 66 11.2 Examples ............................................... 67 11.3 Intersections .............................................. 67 11.4 Disjoint unions and partitions ..................................... 68 11.5 See also ................................................ 68 11.6 References ............................................... 68 11.7 External links ............................................. 69 12 Disjoint union 70 12.1 Example ................................................ 70 12.2 Set theory definition .......................................... 70 12.3 Category theory point of view ....................................