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.

Asked by Abhisek | 1 year ago |  174

##### Solution :-

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

(i) It can be seen that 2 ∈ N; however, 2 ≠ 22 = 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 = 32.

Now, 3 ≠ 92 = 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 = 42 and 4 = 22.

Now, 16 ≠ 22 = 4; therefore, (16, 2) ∉ N

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

Answered by Pragya Singh | 1 year ago

### Related Questions

#### Let R = {(a, b) : a, b, ϵ N and a < b}.Show that R is a binary relation on N

Let R = {(a, b) : a, b, ϵ N and a < b}.Show that R is a binary relation on N, which is neither reflexive nor symmetric. Show that R is transitive.

#### Let A = (1, 2, 3} and B = {4} How many relations can be defined from A to B.

Let A = (1, 2, 3} and B = {4} How many relations can be defined from A to B.

#### Let R = {(x, x2) : x is a prime number less than 10}.

Let R = {(x, x2) : x is a prime number less than 10}.

(i) Write R in roster form.

(ii) Find dom (R) and range (R).