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Question
Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
Solution
(i) Reflexive:
Let A = {1, 2, 3}.
A relation R on A is defined as R = {(1, 2), (2, 1)}.
It is seen that (1, 1), (2, 2), and (3, 3) ∉R.
∴ R is not reflexive.
(ii) Symmetric:
Now, as (1, 2) ∈ R and (2, 1) ∈ R,
∴ R is symmetric.
(iii) Transitive:
Now, (1, 2) and (2, 1) ∈ R
However,
(1, 1) ∉ R
∴ R is not transitive.
Hence, R is symmetric but neither reflexive nor transitive.
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