Equivalence Relation

An Equivalence Relation is a binary relation that is a reflexive relation, a symmetric relation, and a transitive relation.

• AKA: Equivalence Order, ≡, ~.
• Context:
  • It can range from being a Partial Equivalence Relation to being a Full Equivalence Relation.
  • It can range from being a Singleton Equivalence Relation to being a Set Equivalence Relation.
  • It can define an Equivalence Class Set.
  • …
• Example(s):
  • ≡(2,2) ⇒ True.
  • ≡(x,x) ⇒ True.
  • ≡({2},{1,2,3}) ⇒ False.
  • ≡({1,2,3},{1,2,3}) ⇒ True.
  • a Semantic Equivalence Relation.
  • …
• Counter-Example(s):
  • a Proper Subset Relation.
• See: Semantic Equivalence, Identity Relation, if And Only if, Set (Mathematics), Partition of a Set, Empty Set, Clustering Task.

---

---

References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/equivalence_relation Retrieved:2015-6-2.
  • In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of the set is a member of one and only one cell of the partition. The intersection of any two different cells is empty; the union of all the cells equals the original set. These cells are formally called equivalence classes.
  • Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are "a ~ b" and "a ≡ b”, which are used when R is the obvious relation being referenced, and variations of "a ~R b”, "a ≡_R b”, or "aRb" otherwise.
  • A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Equivalently, for all a, b and c in X:
    • a ~ a. (Reflexivity)
    • if a ~ b then b ~ a. (Symmetry)
    • if a ~ b and b ~ c then a ~ c. (Transitivity)
  • X together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted [a], is defined as [math]\displaystyle{ [a] = \{b\in X \mid a\sim b\} }[/math] .

2005

• (ANSI Z39.19, 2005) ⇒ ANSI. (2005). “ANSI/NISO Z39.19 - Guidelines for the Construction, Format, and Management of Monolingual Controlled Vocabularies." ANSI.
  • QUOTE: "equivalence relationship A relationship between or among terms in a controlled vocabulary that leads to one or more terms that are to be used instead of the term from which the cross-reference is made; begins with the word SEE or USE.

---