Question

Let R be a relation from A to A defined by R = {(a, b): a, b ∈ N and a = b²} Are the following true? 

(i) (a, a) ∈ R for all a ∈ A 

(ii) (a, b) ∈ R, implies (b, a) ∈ R 

(iii) (a, b) ∈ R, (b, c) ∈ R, implies (a, c) ∈ R. 

Answer 1

Given R= {(a, b): a,b∈N and a = b²} 

= {(1,1), (2,4), (3, 9), (4, 16),…} 

(i) (a, a)∈ R for all a ∈ N is not true because (2, 2) ∉ R. 

(ii) (a, b) ∈ R implies (b, a) ∈ R is not true, because (2,4) ∈ R but (4,2) ∉ R 

(iii) (a, b) ∈ R, (b, c) ∈ R implies (a,c) ∈ R is not true because (2,4) and (4,16) ∈ R but (2,16) ∉ R.