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.
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.