Q. 324.7( 3 Votes )

Let us define a relation R in R as aRb if a ≥ b. Then R is
A. an equivalence relation

B. reflexive, transitive but not symmetric

C. symmetric, transitive but

D. neither transitive nor reflexive not reflexive but symmetric.

Answer :

Given that, aRb if a ≥ b

Now,

We observe that, a ≥ a since every a ∈ R is greater than or equal to itself.

⇒ a ≥ a ⇒ (a,a) ∈ R ∀ a ∈ R

⇒ R is reflexive.

Let (a,b) ∈ R

⇒ a ≥ b

But b cannot be greater than a if a is greater than b.

⇒ (b,a) ∉ R

For e.g., we observe that (5,2) ∈ R i.e 5 ≥ 2 but 2 ≱ 5 ⇒ (2,5) ∉ R

⇒ R is not symmetric

Let (a,b) ∈ R and (b,c) ∈ R

⇒ a ≥ b and b ≥ c

⇒ a ≥ c

⇒ (a,c) ∈ R

For e.g., we observe that

(5,4) ∈ R ⇒ 5 ≥ 4 and (4,3) ∈ R ⇒ 4 ≥ 3

And we know that 5 ≥ 3 ∴ (5,3) ∈ R

⇒ R is transitive.

Thus, R is reflexive, transitive but not symmetric.

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Let us define a relation R in R as aRb if a ≥ b. Then R is A. an equivalence relationB. reflexive, transitive but not symmetricC. symmetric, transitive butD. neither transitive nor reflexive not reflexive but symmetric.