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Question

a) Show that congruence modulo m is an equivalence relation whenever m is a positive integer. b) Show that the relation {(a, b) | a ≡ ±b (mod 7)} is an equivalence relation on the set of integers.

Solution

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1 of 6DEFINITIONS

A relation $R$ is an $\textbf{equivalence relation}$ if the relation $R$ is transitive, symmetric and reflexive.

A relation $R$ on a set $A$ is $\textbf{reflexive}$ if $(a,a)\in R$ for every element $a\in A$.

A relation $R$ on a set $A$ is $\textbf{symmetric}$ if $(b,a)\in R$ whenever $(a,b) \in R$

A relation $R$ on a set $A$ is $\textbf{transitive}$ if $(a,b)\in R$ and $(b,c) \in R$ implies $(a,c)\in R$

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