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Question
Test whether the following relation R3 is (i) reflexive (ii) symmetric and (iii) transitive:
R3 on R is defined by (a, b) ∈ R3 `⇔` a2 – 4ab + 3b2 = 0.
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Solution
Reflexivity:-
Let a be an arbitrary element of R3.
Then, a ∈ R3
⇒ a2 − 4a × a + 3a2 = 0
So, R3 is reflexive.
Symmetry:-
Let (a, b) ∈ R3
⇒ a2 − 4ab + 3b2 = 0
But b2 − 4ba + 3a2 ≠ 0 for all a, b ∈ R
So, R3 is not symmetric.
Transitivity:-
Let (1, 2) ∈ R3 and (2, 3) ∈ R3
⇒ 1 − 8 + 6 = 0 and 4 − 24 + 27 = 0
−7 + 6 = 0 and −20 + 27 = 0
−1 ≠ 0 and 7 ≠ 0
So, R3 is not transitive.
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