5.14 Equivalence relations
equivalence relation, A relation R is an equivalence relation if R is reflexive, symmetric, and transitive
equivalence class, the set [a]
partition, is a set of non-empty subsets of A that are pairwise disjoint and whose union is A.
pairwise disjoint, A set of sets is pairwise disjoint if the intersection of any pair of the sets is empty
overview:
If R is reflexive, symmetric, and transitive, it is an equivalence relation. The notation ab is used to indicate aRb when a relation R is an equivalence relation. "A is equivalent to b," reads ab. If A is the domain of an eguivalence relation and an A is the set of all x A such that ax,then [a] is defined as the set of all x A such that ax. A set [a] is referred to as an equivalence class. A partition of a set A is a collection of non-empty, pairwise disjoint subsets of A whose union is A.
Example of Equivalence relation
A={a,b,c,d,e,f} R on A
R is an equivalence relation: Reflexive Symmetric Transitive (e,a)and (a,b)>(e,b) Add (e, d) and no longer transitive: (b, e) and (e, d) but not (b, d)
Captions
1. R is reflexive (all self-loops are present) 2. R is symmetric (if (x,y) is an edge, then (y,x) is also an edge) 3. R is transitive (edges (e,a) and (a,b) imply the presence of edge (e,b)). Therefore, R is an equivalence relation. 4. If an edge (e,d) is added, then R is no longer transitive: (b,e) and (e,d) are edges, but there is no edge from b to d.
Theorem 5.14.2: Equivalence relations define a partition,
Assuming that there's an equivalence relation over a set A. The set of all distinct equivalence classes defines a partition of A. The term "distinct" means that if there are two equal equivalence classes [a] = [b], the set [a] is only included once