If R be a relation on the set of integers Z, given by R = {(a, b) : (a - b) is a multiple of 3}, then R is:

R is a relation on integers Z given by R = {(a, b) : (a - b) is a multiple of 3}

Two numbers are related if their difference is divisible by 3.

For R to be an equivalence relation, it must satisfy three properties: reflexive, symmetric, and transitive.

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For reflexivity, check if (a, a) is in R.

$(a - a) = 0$

Since $0 = 3 \times 0$, zero is a multiple of 3.

Therefore, R is reflexive.

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For symmetry, assume (a, b) is in R, meaning $(a - b)$ is a multiple of 3.

$(a - b) = 3k$ for some integer $k$

$b - a = -(a - b)$

$b - a = -3k$

$b - a = 3(-k)$

Since $-k$ is an integer, $(b - a)$ is a multiple of 3.

Therefore, R is symmetric.

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For transitivity, assume (a, b) is in R and (b, c) is in R.

$a - b = 3k$ for some integer $k$

$b - c = 3m$ for some integer $m$

$a - c = (a - b) + (b - c)$

$a - c = 3k + 3m$

$a - c = 3(k + m)$

Since $(k + m)$ is an integer, $(a - c)$ is a multiple of 3.

Therefore, R is transitive.

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Since R is reflexive, symmetric, and transitive, R is an equivalence relation.