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Question
S is a relation over the set R of all real numbers and it is given by (a, b) ∈ S ⇔ ab ≥ 0. Then, S is _______________ .
Options
symmetric and transitive only
reflexive and symmetric only
antisymmetric relation
an equivalence relation
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Solution
an equivalence relation
Reflexivity: Let a ∈ R
Then,
aa = a2 > 0
⇒ (a, a) ∈ R ∀ a ∈ R
So, S is reflexive on R.
Symmetry: Let (a, b) ∈ S
Then,
(a, b) ∈ S
⇒ ab ≥ 0
⇒ ba ≥ 0
⇒ (b, a) ∈ S ∀ a, b ∈ R
So, S is symmetric on R.
Transitive:
If (a, b), (b, c) ∈ S
⇒ ab ≥ 0 and bc ≥ 0
⇒ ab x bc ≥ 0
⇒ ac ≥ 0 [∵ b2 ≥ 0]
⇒ (a, c) ∈ S for all a, b, c ∈ set R
Hence, S is an equivalence relation on R
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