Question

Let R be a relation from Q to Q defined by R={(a,b): a,b belongs to Q and a-b belongs to z}, show that, (i) (a,a) belongs to R for all a belongs to Q (ii) (a,b) belongs to R implies that (b,a) belongs to R. (iii) (a,b) belongs to Road and (be, c) belongs to R implies that (a,c) belongs to R.

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Answer:

(i) Here,R = {(a,b):a,b ∈ Q and a−b ∈ Z}

(a,a) = a−a = 0 ∈ Z

So, (a,a) ∈ R for all a ∈ Q

(ii)We are given, (a−b) ∈ Z⇒ −(a−b) ∈ Z ⇒ (b−a) ∈ Z So,(a,b) ∈ R implies (b,a)∈R

(iii) we are given,

(a−b)∈Z and (b−c)∈Z

So,(a−b)+(b−c)∈Z⇒(a−c)∈Z

Thus,(a,b) ∈ R and (b,c)∈R implies that (a,c) ∈ R