#### Question

Sum

Show that the relation R on the set Z of all integers, given by R = {(a,b) : 2 divides (a-b)} is an equivalence relation.

#### Solution

To prove relation is an equivalence relation

We have to show three properties

1. Reflexive

(a,a) ∈ R

2. Symmetric

(a,b) ∈ R

⇒ (b,a) ∈ R

3. Transitive

(a,b) ∈ R and (b,c) ∈ R

⇒ (a,c) ∈ R

1. R is reflexive because 2 divides (a - a)∀a ∈ z∀a ∈ z

2. 2 divides a - b

therefore, 2 divides b - a hence, (b,a) ∈ R

R is symmetric

3. (a, b) ∈ R

(b, c) ∈ R

then a − b and b − c are divisible by 2.

Now, a − c = ( a − b ) + ( b − c) = a−c

so, a − c is divisible by 2.

Therefore, (a, c ) ∈ R

Therefore, R is an equivalence relation.

Concept: Types of Relations

Is there an error in this question or solution?

Advertisement Remove all ads

#### APPEARS IN

Advertisement Remove all ads

Advertisement Remove all ads