Question

let R be a relation from N to N defined by R= {(a, b): a, b EN & a=b²} Are the following true?
(i) (a, a) ER, for all a EN
(ii) ca, b) ER, implies (b, a) ER
(iii) (a, b) ER, (b, c) ER implies (a, c) ER. Justify your answer in each case​

Answer 1

Answer:

(i) (a, a) ER, for all a EN

(ii) ca, b) ER, implies (b, a) ER

(iii) (a, b) ER, (b, c) ER implies (a, c) ER. Justify your answer in each case

Answer 2

Answer:

R = {(a, b): a, b ∈ N and a = b2} (i) It can be seen that 2 ∈ N; however, 2 ≠ 22 = 4. Therefore, the statement “(a, a) ∈ R, for all a ∈ N” is not true. (ii) It can be seen that (9, 3) ∈ N because 9, 3 ∈ N and 9 = 32. Now, 3 ≠ 92 = 81; therefore, (3, 9) ∉ N Therefore, the statement “(a, b) ∈ R, implies (b, a) ∈ R” is not true. (iii) It can be seen that (9, 3) ∈ R, (16, 4) ∈ R because 9, 3, 16, 4 ∈ N and 9 = 32 and 16 = 42. Now, 9 ≠ 42 = 16; therefore, (9, 4) ∉ N

Therefore, the statement “(a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R” is not true.