# Prove that the relation R on Z, defined by R {(x, y) : (x – y) is divisible by 5} is an equivalence relation.

Answers (1)

R on Z and R = {(x, y) : (x– y) is divisible by 5}

$\\ Check for Reflexive: \\ if (\mathrm{a}, \mathrm{a}) for every a \in \mathrm{R} \Rightarrow \mathrm{R} is reflexive \\ Here (a-a) divisible by 5 \Rightarrow (\mathrm{a}, \mathrm{a}) \in \mathrm{R} \\ \mathrm{R} is reflexive \\$

$\\ Check for Symmetric: \\ if (\mathrm{a}, \mathrm{b}) \in \mathrm{R} then (\mathrm{b}, \mathrm{a}) \in \mathrm{R} \Rightarrow \mathrm{R} is Symmetric$

$\\ Here (a-b) is divisible by 5 \Rightarrow (a-b) = 5 k \\ (b-a) = -5k \Rightarrow (b-a) is divisible by 5 \\ R is symmetric$

$\\ Check For transitive : \\ If (a, b),(b, c) \in R then (a, c) \in R \Rightarrow R is transitive$

$\\ (a-b)= 5k, (b-c) = 5l \\ (a-c )= (a-b)+(b-c) = 5k +5l \\ (a-c) = 5(k+l) \Rightarrow (a-c) is divisible by 5 \\ R is transitive$

Hence R an equivalence relation.

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