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.