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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 ...
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]
The identity element is the empty relation. ... is parallel to " is an equivalence relation on the set of all lines in the Euclidean plane. All operations ...
Empty relation E = ∅; that is, x 1 Ex 2 ... The number of equivalence relations is the number of partitions, which is the Bell number. The homogeneous relations can ...
If and are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function : is a homomorphism, then is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of is a quotient of . [2] The bijection between the coimage and the image of ...
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the ...
The empty relation on any set is transitive [3] ... Strict weak ordering – a strict partial order in which incomparability is an equivalence relation;
2. An equivalence relation \ The difference of two sets: x\y is the set of elements of x not in y. − The difference of two sets: x−y is the set of elements of x not in y. ≈ Has the same cardinality as × A product of sets / A quotient of a set by an equivalence relation ⋅ 1. x⋅y is the ordinal product of two ordinals 2.