Question

Show that the relation R in R defined as R = {(a, b): a ≤ b} , is reflexive and transitive
but not symmetric.

Answer 1

R = {(a, b); a ≤ b}
Clearly (a, a) ∈ R as a = a.
∴R is reflexive.

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Now,
(2, 4) ∈ R (as 2 < 4)
But, (4, 2) ∉ R as 4 is greater than 2.
∴R is not symmetric.

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Now, let (a, b), (b, c) ∈ R.
Then,
a ≤ b and b ≤ c
⇒ a ≤ c
⇒ (a, c) ∈ R
∴R is transitive.

**Hence, R is reflexive and transitive but not symmetric. **