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Question
The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is ___________________ .
Options
symmetric only
reflexive only
an equivalence relation
transitive only
Solution
an equivalence relation
R={ (a, b) : a = b and a, b ∈ A }
Reflexivity : Let a ∈ A
Here,
a = a
⇒ (a, a) ∈ R for all a ∈ A
So, R is reflexive on A
Symmetry : Let a, b ∈ A such that (a, b) ∈ R. Then,
(a, b) ∈ R
⇒ a = b
⇒ b = a
⇒ (b, a) ∈ R for all a ∈ A
So, R is symmetric on A.
Transitive : Let a, b, c ∈ A such that (a, b) ∈ R and (b, c) ∈ R.
Then, (a, b) ∈ R ⇒ a = b and (b, c) ∈ R ⇒ b = c
⇒ a = c
⇒ (a, c) ∈ R for all a ∈ A
So, R is transitive on A.
Hence, R is an equivalence relation on A.
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