Let R be a relation define as R = {(a, b): a2 ≥ b, where a and b ∈

1.	Let R be a relation define as R = {(a, b): a2 ≥ b, where a and b ∈ Z} . Then relation R is a/an 1. Reflexive and symmetric2. Symmetric and transitive3. Transitive and reflexive4. Reflexive only
Answer» Correct Answer - Option 4 : Reflexive only Concept: A relation R in a set A is called (i) Reflexive, if (a, a) ∈ R, for every a ∈ A. (ii) Symmetric, if (a, b) ∈ R implies that (b, a) ∈ R, for all a, b ∈A. (iii) Transitive, if (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R, for all a, b, c ∈A. Calculation: Given: R = {(a, b): a2 ≥ b} We know that a2 ≥ a Therefore (a, a) ∈ R, for all a ∈ Z. Hence, relation R is reflexive. Let (a, b) ∈ R ⇒ a2 ≥ b but b2 \(\ngeqslant\) a for all a, b ∈ Z. So, if (a, b) ∈ R then it does not implies that (b, a) also belongs to R. Hence, relation R is not symmetric. Now let (a, b) ∈ R and (b, c) ∈ R. ⇒ a2 ≥ b and b2 ≥ c This does not implies that a2 ≥ c, therefore (a, c) does not belong to R for all a, b, c ∈ Z. Hence, relation R is not transitive. Hence, option 4 is the correct answer.

Let R be a relation define as R = {(a, b): a2 ≥ b, where a and b ∈ Z} . Then relation R is a/an 1. Reflexive and symmetric2. Symmetric and transitive3. Transitive and reflexive4. Reflexive only

Answer» Correct Answer - Option 4 : Reflexive only

Concept:

A relation R in a set A is called

(i) Reflexive, if (a, a) ∈ R, for every a ∈ A.

(ii) Symmetric, if (a, b) ∈ R implies that (b, a) ∈ R, for all a, b ∈A.

(iii) Transitive, if (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R, for all a, b, c ∈A.

Calculation:

Given: R = {(a, b): a2 ≥ b}

We know that a2 ≥ a

Therefore (a, a) ∈ R, for all a ∈ Z.

Hence, relation R is reflexive.

Let (a, b) ∈ R

⇒ a2 ≥ b but b2 \(\ngeqslant\) a for all a, b ∈ Z.

So, if (a, b) ∈ R then it does not implies that (b, a) also belongs to R.

Hence, relation R is not symmetric.

Now let (a, b) ∈ R and (b, c) ∈ R.

⇒ a2 ≥ b and b2 ≥ c

This does not implies that a2 ≥ c, therefore (a, c) does not belong to R for all a, b, c ∈ Z.