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This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1. The empty relation R (defined so that aRb is never true) on a set X is vacuously symmetric and transitive; however, it is not reflexive (unless X itself is empty).
The initial definition of a cardinal number is an equivalence class of sets, where two sets are equivalent if there is a bijection between them. The difficulty is that almost every equivalence class of this relation is a proper class , and so the equivalence classes themselves cannot be directly manipulated in set theories, such as Zermelo ...
In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion. A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice.
There are four different equivalence relations which may be defined on the set of functions f from N to X: equality; equality up to a permutation of N; equality up to a permutation of X; equality up to permutations of N and X. The three conditions on the functions and the four equivalence relations can be paired in 3 × 4 = 12 ways.
The cardinality of a set X is essentially a measure of the number of elements of the set. [1] Equinumerosity has the characteristic properties of an equivalence relation (reflexivity, symmetry, and transitivity): [1] Reflexivity Given a set A, the identity function on A is a bijection from A to itself, showing that every set A is equinumerous ...
In set theory, the kernel of a function (or equivalence kernel [1]) may be taken to be either the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function can tell", [2] or; the corresponding partition of the domain.
Conditions 1, 2, and 3 say that ~ is an equivalence relation. A congruence ~ is determined entirely by the set {a ∈ G | a ~ e} of those elements of G that are congruent to the identity element, and this set is a normal subgroup. Specifically, a ~ b if and only if b −1 * a ~ e.