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Question
If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is ____________ .
Options
reflexive
symmetric
transitive
all the three options
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Solution
all the three options
R={ (a, b) : a=b and a, b ∈ A }
Reflexivity: Let a ∈ A. Then,
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.
Transitivity : 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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