Q. 1D3.5( 120 Votes )

# Relation R in the

Answer :

It is given that Relation R in the set Z of all integers defined as

R = {(x, y) : x – y is an integer}

Now, for every x ϵ Z, (x, x) ϵ R, as x –x = 0 is an integer.

⇒ R is reflexive.

Now, for every x, y ϵ Z if (x, y) ϵ R, then x – y is an integer.

⇒ -(x – y) is also an integer.

⇒ (y – x) is an integer.

⇒ (y, x) ϵ R

⇒ R is symmetric.

Now, for every (x, y) and (y, z) ϵ R where x, y, z ϵ R,

⇒ (x – y) and (y – z) is an integer.

⇒ (x – z) = (x – y) + (y – x) is an integer.

⇒ (x, z) ϵ R

⇒ R is transitive.

Therefore, R is reflexive, symmetric and transitive.

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