Week 5 note
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This 1 page Class Notes was uploaded by Zuizui on Sunday September 25, 2016. The Class Notes belongs to Math 301 at University of Mississippi taught by Dr. Laura Sheppardson in Fall 2016. Since its upload, it has received 3 views. For similar materials see Discrete Mathematics in Math at University of Mississippi.
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Date Created: 09/25/16
An equivalence relation on a set X is a subset of X×X, i.e., a collection R of ordered pairs of elements of X, satisfying certain properties. Write "xRy" to mean (x,y) is an element of R, and we say "x is related to y," then the properties are 1. Reflexive: aRa for all a in X, 2. Symmetric: aRb implies bRa for all a,b in X 3. Transitive: aRb and bRc imply aRc for all a,b,c in X, where these three properties are completely independent. Other notations are often used to indicate a relation, e.g., a=b or a∼b. Let R be a relation on a set S. Then R is called • Reflexive if for all x ∈ S we have (x, x) ∈ R; • Irreflexive if for all x ∈ S we have (x, x) 6∈ R; • Symmetric if for all x, y ∈ S we have xRy implies yRx; • Antisymmetric if for all x, y ∈ S we have that xRy and yRx implies x = y; • Transitive if for all x, y, z ∈ S we have that xRy and yRz implies xRz.
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