2.7.4 Equivalence Relations Video
Equivalence relations are another kind of binary relation on a set which play a crucial role in mathematics and in computer science in particular. And they can also be explained both in terms of digraphs and in terms of axioms. So let's begin with a digraph explanation of an equivalence relation. And the kind of relation that's an equivalence relation is the relation of there being a walk in both directions between two vertices. So if there's a walk between vertex u and vertex v and conversely there's a walk from vertex v back to vertex u, then u and v are said to be strongly connected, and strongly connected is going to be an example of an equivalence relation. So in terms of the walk relation, including 0-length walks, the relation we're talking about is u G* v and v G* u. Now, as a property of relations, this has a name. It's called symmetry. So a relation R on a set A is symmetric if and only if a R b implies b R a, and the first remark is that...