Question

Let A = {1, 2, 3} and R = {(a, b) : a, b ϵ A and |a² – b² | ≤ 5. 

Write R as a set of ordered pairs.

Mention whether R is (i) reflexive (ii) symmetric (iii) transitive. Give reason in each case.

Answer 1

Put a = 1 , b = 1 |1² – 1² | ≤ 5, (1, 1) is an ordered pair.

Put a = 1 , b = 2 |1² – 2² | ≤ 5, (1, 2) is an ordered pair. 

Put a = 1 , b = 3 |1² – 3² | > 5, (1, 3) is not an ordered pair. 

Put a = 2 , b = 1 |2² – 1² | ≤ 5, (2, 1) is an ordered pair.

Put a = 2 , b = 2 |2² – 2² | ≤ 5, (2, 2) is an ordered pair. 

Put a = 2 , b = 3 |2² – 3² | ≤ 5, (2, 3) is an ordered pair. 

Put a = 3 , b = 1 |3² – 1² | > 5, (3, 1) is not an ordered pair. 

Put a = 3 , b = 2 |3² – 2² | ≤ 5, (3, 2) is an ordered pair. 

Put a = 3 , b = 3 |3² – 3² | ≤ 5, (3, 3) is an ordered pair. 

R = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2), (3, 3)}

(i) For (a, a) є R

|a² – a² | = 0 ≤ 5. Thus, it is reflexive.

(ii) Let (a, b) є R 

(a, b) є R è |a² – b² | ≤ 5

|b² – a² | ≤ 5 

(b, a) є R 

Hence, it is symmetric 

(iii) Put a = 1 , b = 2 , c = 3.

|1² – 2² | ≤ 5

|2² – 3² | ≤ 5 

But |1² – 3²| > 5 

Thus, it is not transitive.