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The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest. The relation " ∼ {\displaystyle \sim } is finer than ≈ {\displaystyle \approx } " on the collection of all equivalence relations on a fixed set is itself a partial order ...
Let be the set of ordered pairs of integers (,) with non-zero , and define an equivalence relation on such that (,) (,) if and only if =, then the equivalence class of the pair (,) can be identified with the rational number /, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of ...
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.
The partitions of a set correspond one-to-one with its equivalence relations. These are binary relations that are reflexive , symmetric , and transitive . The equivalence relation corresponding to a partition defines two elements as being equivalent when they belong to the same partition subset as each other.
For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from any partition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the same part in P. Thus the notions of equivalence relation and partition are essentially equivalent. [5]
Conversely, every partition defines an equivalence class. The equivalence relation of equality is a special case, as, if restricted to a given set , it is the strictest possible equivalence relation on ; specifically, equality partitions a set into equivalence classes consisting of all singleton sets.
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–Fraenkel set theory, that only deal with sets. It is often desirable in the context of set theory to have sets that are representatives for the ...
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.