Question

Check whether the relation r in the set r of real numbers, defined by r = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive.

Answer 1

reflexive : A relation is said to be reflexive if (x, x)  ϵ  R, where x is from domain. 

relation = {(a, b) , 1 + ab > 0 where a , b ϵ R }
for any a ϵ R , a² ≥ 0
so, 1 + a² ≥ 1
or , 1 + a² > 0
⇒ 1 + a.a > 0
⇒(a , a) ϵ R
therefore, Relation is reflexive


symmetirc : A relation is said to be symmetric if (y, x)  ϵ R whenever (x, y) ϵ R.

Relation = {(a, b) , 1 + ab > 0 where a , b ϵ R }
Let (a, b) ϵ R where a, b ϵ R
so, 1 + ab > 0
⇒ 1 + ba > 0
⇒(b, a) ϵ R
therefore, Relation is symmetric .

transitive : A relation is said to be transitive if (x, z)  ϵ  R whenever (x, y) ϵ R and (y, z) ϵ R.

Relation = {(a, b) , 1 + ab > 0 where a , b ϵ R }
Let's take (-2, 0) and (0, 1) such that, 1 + -2 × 0 > 0 and 1 + 0 × 1 > 0
so, (-2, 0) and (0, 1) belongs to R.
but (-2, 1) doesn't belong to R because 1 + -2 × 1 < 0
this doesn't follow transitive.
hence, Relation is not transitive.

Answer 2

This is the answer.

Last line is not transitive