If R is a relation on the set of integers Z such that R = {(a, b) : 2 divides a - b where a, b ∈ Z} then R is a/an ?

Concept:

• Reflexive:

Let R be a relation on a non-empty set A, if every element of A is related to itself then R is said to be a reflexive relation.

Thus, R is reflexive ⇔ (a, a) ∈ R, ∀ a ∈ A.

• Symmetric:

Let R be a relation on a non-empty set A, then the relation R is said to be symmetric relation ⇔ (a, b) ∈ R ⇒ (b, a) ∀ a, b ∈ A.

• Transitive:

Let R be a relation on a non-empty set A, then the relation R is said to be transitive relation ⇔ (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R, ∀ a, b, c ∈ A.

• Equivalence:

Let R be a relation on a non-empty set A, then the relation R is said to be equivalence relation if R is reflexive, symmetric and transitive.

Calculation:

Given: R is a relation on the set of integers Z such that R = {(a, b) : 2 divides a - b where a, b ∈ Z}

Reflexive:

As we can see that a - a = 0 ∀ a ∈ Z ⇒ (a, a) ∈ R ∀ a ∈ Z

So, the given relation R is reflexive.

Symmetric:

Let (a, b) ∈ R ⇒ 2 divides a - b i.e a - b = 2k where k is some integer

So, if a - b = 2k ⇒ b - a = 2l where l = - k ∈ Z

So, 2 also divides b - a ⇒ (b, a) ∈ R

So, the given relation R is  also symmetric.

Transitive:

Let (a, b) and (b, c) ∈ R i.e 2 divides both a - b and b - c also.

⇒ a - b = 2k where k is some integer and similarly b - c = 2l where l is some integer.

⇒ (a - b) + (b - c) = 2k + 2l

⇒ a - c = 2m where m = l + k ∈ Z

So, 2 also divides a - c ⇒ (a, c) ∈ R

So, the given relation R is also transitive.

As we know that if a relation R is reflexive, symmetric and transitive then R is said to be an equivalence relation

Hence, option 4 is the correct answer.