Question

Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b2}. Are the following true?
(i) (a, a) ∈ R, for all a ∈ N
(ii) (a, b) ∈ R, implies (b, a) ∈ R
(iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R.
Justify your answer in each case.
​

Answer 1

━━━━━━━━━━━━━━━━━━━━━━━━━

$\Large\fbox{\color{purple}{QUESTION}}$

Let R be a relation from N to N defined by R =

{(a, b): a, b ∈ N and a = b2}. Are the following true?

(i) (a, a) ∈ R, for all a ∈ N

(ii) (a, b) ∈ R, implies (b, a) ∈ R

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

Justify your answer in each case.

━━━━━━━━━━━━━━━━━━━━━━━━━

$\bf\Huge\red{\mid{\overline{\underline{ ANSWER }}}\mid }$

➡️Given relation R = {(a, b): a, b ∈ N and a = b^2}

✴(i) It can be seen that 2 ∈ N;

➡️however, 2 ≠ 2^2 = 4.

➡️Thus, the statement “(a, a) ∈ R, for all a ∈ N” is not true.

✴(ii) Its clearly seen that (9, 3) ∈ N because 9, 3 ∈ N and 9 = 3^2.

➡️Now, 3 ≠ 9^2 = 81; therefore, (3, 9) ∉ N

➡️Thus, the statement “(a, b) ∈ R, implies (b, a) ∈ R” is not true.

✴(iii) Its clearly seen that (16, 4) ∈ R, (4, 2) ∈ R because 16, 4, 2 ∈ N and 16 = 4^2 and 4 = 2^2.

➡️Now, 16 ≠ 2^2 = 4; therefore, (16, 2) ∉ N

➡️Thus, the statement “(a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R” is not true.

$\bf\Large\red{ THANKS \: FOR \: YOUR}$

$\bf\Large\red{ QUESTION \: HOPE \: IT }$

$\bf\Large\red{ HELPS }$

$\Large\mathcal\green{FOLLOW \: ME}$

$\Large\mathcal\green{FOLLOW \: ME}$━━━━━━━━━━━━━━━━━━━━━━━━━

Answer 2

Answer:

Given,

R:N------->N defined by

R = {(a,b):a,b∈N and a = b^2}

(i) it's not true . Becoz a = a^2

a(a -1) = 0 , a= 0, 1 hence, it is true only for a = 1 not for other natural number.

(ii) it's not true. I mean statement is false.because if a=b^2, then b=a^2 is not correct statement .

For example,

4 = 2^2 but 4^2 >2

(iii) it's flase. Because

if (a,b)∈R, (b,c)∈R and (a,c)∈R then, it means a=b^2 and b=c^2 , a=c^2

This means a=(c^2)^2= c^4

But a= c^2 from above.

So, statenent is wrong.