The relation "congruence modulo m" | vedclass

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Question

If $R$ is a relation, $x R y \Leftrightarrow x-y$ is divisible by $m$.

$x R x$ because $x-x$ is divisible by $m$.

It is reflexive.

$xRy \Righ

Vedclass · Accepted Answer

An equivalence relation

Vedclass · Answer

If $R$ is a relation, $x R y \Leftrightarrow x-y$ is divisible by $m$.

$x R x$ because $x-x$ is divisible by $m$.

It is reflexive.

$xRy \Rightarrow x - y$ is divisible by $m$.

$y - x$ is divisible by $m$.

$y yx$

It is symmetric.

$x R y$ and $y R z$

$x-y=k_1 m, y-z=k_2 m$

$x-z=\left(k_1+k_2\right) m$

It is transitive.

Since $R$ is reflexive, symmetric and transitive, it is an equivalence relation.

The relation "congruence modulo $m$" is

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