For the relation R 1 defined on R by the rule (a, b) R 1 1 + ab 0. Prove that: (a, b) R 1 and (b,c) R 1 (a, c) R 1 is not true for all a, b, c R.

Question

For the relation R 1 defined on R by the rule (a, b) R 1 1 + ab > 0. Prove that: (a, b) R 1 and (b,c) R 1 (a, c) R 1 is not true for all a, b, c R.

Philoid · Accepted Answer

To prove: (a, b) R₁ and (b,c) R₁ (a, c) R₁ is not true for all a, b, c R.

Given R₁ = {(a, b): 1 + ab > 0}

Let a = 1, b = -0.5, c = -4

Here, (1, -0.5) R₁ [∵ 1+(1×-0.5) = 0.5 > 0]

And, (-0.5, -4) R₁ [∵ 1+(-0.5×-4) = 3 > 0]

But, (1, -4) ∉ R₁ [∵ 1+(1×-4) = -3 < 0]

∴ (a, b) R₁ and (b,c) R₁ (a, c) R₁ is not true for all a, b, c R

Hence Proved.

NOTE:

Here R₁ is a relation whereas R denotes a real number.