Let R be a relation from N to N defined by R= {(a, b): a, b N and a = b2}.

Are the following statements true?


i. (a, a) R for all a N


ii. (a, b) R (b, a) R


iii. (a, b) R and (b, c) R (a, c) R

Given, R= {(a, b): a, b N and a = b2}

i. (a, a) R for all a N


Here, take b = 2


a = b2 = 22 = 4


(4, 2) R but (2, 2) R


As, 22 ≠ 2


So,


No, the statement is false.


ii. (a, b) R (b, a) R


Here, take b = 2


a = b2 = 22 = 4


(4, 2) R but (2, 4) R


As, 42 ≠ 2


So,


No, the statement is false.


iii. (a, b) R and (b, c) R (a, c) R


Here, take b = 4


a = b2 = 42 = 16


(16, 4) R


Now, b = c2


4 = c2


c = -2 N or 2 N


(4, 2) R


But (16, 2) R


As, 22 ≠ 16


So,


No, the statement is false.


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