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
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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
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.