Equivalence Classes

MATH 321 Equivalence RELATIONS,WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THERATIONAL NUMBERSALLAN explore the notion of well-definedness when defining functionswhose domain is the set of all Equivalence Classes of an Equivalence we apply this to define modular arithmetic and the setQof ClassesWe shall slightly adapt our notation for relations in this document. Let bea relation on a setX. Formally, is a subset ofX X. Given two elementsx,y X, we shall writex yto mean (x,y) . From now on, we shall just usethe notationx y, and not explicitly reference as a subset ofX that a relation on a setXis anequivalence relationif is(i)reflexive: ( x X)(x x),(ii)symmetric: ( x,y X)(x y y x), and(iii)transitive: ( x,y,z X)((x y y z) (x z)).Suppose is an Equivalence relation onX. When two elements are related via , it is common usage of language to say they areequivalent.

x2X, prove existence and uniqueness of z2Zfor which (x;z) 2g fseparately. To prove uniqueness, suppose (x;z 1);(x;z 2) 2g f, and show that z 1 = z 2.) EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS 5 We can translate the de nitions of injectivity and surjectivity in terms of the set f. De nition. Let f X Y be a function.

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  Existence, Uniqueness, Existence and uniqueness

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