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1.Relation and Function
normal
The સંબંધ "congruence modulo $m$" is
A
Reflexive only
B
પરંપરિત only
C
સંમિત only
D
An equivalence સંબંધ
Solution
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.
Standard 12
Mathematics
