Q. 93.7( 54 Votes )

# Let R be a relation from N to N defined by R = {(a, b): a, b, ε N and a = b ^{2}}. Are the following true?

(i) (a,a) ε R, for all a ε N

(ii) (a,b) ε R, implies (b,a) ε R

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

Justify your answer in each case.

Answer :

Given: R = {(a,b): a, b Є N and a = b^{2}}

(i) (a,a) Є R, for all a Є N

As we can see that 3 Є N but 3 ≠ 3^{2} = 9.

Hence, the statement is not true.

(ii) (a,b) Є R, implies (b,a) Є R

As we can see that (4,2) Є N and 4 = 2^{2} = 4 but 2 ≠ 4^{2} = 16 hence, (2,4) does not belong to N.

Hence, the statement is not true.

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

As we see, (9, 3) Є R, (16, 4) Є R because 3,4,9,16 Є N and 9 = 3^{2} and 16 = 4^{2}.

Now, 9 ≠ 4^{2} = 16; therefore, (9, 4) does not belong to N.

Hence, the statement is not true.

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