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.
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.