Question

Prove that the relation r on the set a ={1,2,3,4,5} given by r

Answer 1

Given the set A={1,2,,3,4,5} and the relation R={(a,b):|a−b|iseven}:

Let a=b, (a,a)∈R→|a−a|=0 which is even. Therefore R is reflexive.

For R to be symmetric, if (a,b)∈R→(b,a)∈R.

(a,b)∈R→|a−b|=even

(b,a)∈R→|b−a|=even

(a−b)=−(b−a); therefore |(a−b)|=|(b−a)|

Therefore (b,a)∈R. Hence, R is symmetric

Let (a,b)∈Rand(b,c)∈R

⇒|a−b|iseven and |b−c|iseven

If (a,c)∈R→|a−c|=even

Now, a−c=a−b+b−c, which is an even number, as the sum of even numbers is even.

Hence R is transitive.

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