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Question
For the relation R1 defined on R by the rule (a, b) ∈ R1 ⇔ 1 + ab > 0. Prove that: (a, b) ∈ R1 and (b , c) ∈ R1 ⇒ (a, c) ∈ R1 is not true for all a, b, c ∈ R.
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Solution
We have:
(a, b) ∈ R1 ⇔ 1 + ab > 0
Let:
a = 1, b = \[- \frac{1}{2}\]and c = -4
Now,
\[\left( 1, - \frac{1}{2} \right) \in R_1 \text{ and } \left( - \frac{1}{2}, - 4 \right) \in R_1 \] , as
(a, b) ∈ R1 and (b , c) ∈ R1
Thus, (a, c) ∈ R1 is not true for all a, b, c ∈ R.
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