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

1.	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
Answer» 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.

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

Answer»

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.

Discussion

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