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Question
Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
Solution
(i) Reflexive:
R = {(a, b); a ≤ b}
Clearly, (a, a) ∈ R as a = a.
∴R is reflexive.
(ii) Symmetric:
Now, (2, 4) ∈ R (as 2 < 4)
But, (4, 2) ∉ R, as 4 is greater than 2.
∴ R is not symmetric.
(iii) Transitive:
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.
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