Check if the binary relation R defined over Z such that (x, y) ∈ R if and only if x − y is divisible by 4 is an equivalence relation. Ex

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R on Z= {(x, y) ∈ R if and only if x − y is divisible by 4}

Reflexive:

(a - a) = 0 is divisible by 4.

So, (a,a) ∈ R

$\therefore$ R is reflexive.

Symmetric:

(a-b) is divisible by 4.

$\Rightarrow$ -(b-a) is divisible by 4.

$\Rightarrow$ (b-a) is divisible by 4.

$\therefore$ R is symmetric.

Transitive:

(a-b) is divisible by 4, (b-c) is divisible by 4.

Then, (a-b)+(b-c) is divisible by 4.

$\Rightarrow$ (a-b+b-c) is divisible by 4.

$\Rightarrow$ (a-c) is divisible by 4.

$\therefore$ R is transitive.

Thus, R is an equivalence relation on Z.