MATH 213 Lecture Notes - Equivalence Class, Antisymmetric Relation

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A directed graph, consists of a set v of vertices (or nodes) together with a set e of ordered pairs of elements of v called edges (or arcs). The vertex a is called the initial vertex of the edge (a, b), and the vertex b is called the terminal vertex of this edge. An edge of the form (a, a) is called a loop. R = {(a, b), (b, a), (b, b), (c, b), (a, d)} Symmetric: (a, b), (b, a) in the opposite direction. Anti-symmetric: edge (a, b), a != b, edge (b, a) in the opposite direction. Transitive: (a, b) (b, c), edge (a, c) A relation on a set a is an equivalence relation if it is reflexive, symmetric, and transitive. If so, then arb is denoted a ~ b. Then, a ~ a a ~ b b ~ a a ~ b b ~ c a ~ c.

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