Which of the following sets are closed under the binary operation $*$ defined as: $a*b=\frac{a+b}{1+ab}$

Question

Which of the following sets are closed under the binary operation $*$ defined as: $$a*b=\frac{a+b}{1+ab}$$

$1.\{x\in \mathbb{R}:x\geq 0\}$

$2.\{x\in \mathbb{R}:|x|>1 \}$

$3.\{x\in \mathbb{R}:|x|<1 \}$

I have proved that first is closed.

Second is not closed if we take $a=b=2$

About third I am unable to prove or disprove it, need help about the last option.

Answer 1

As a hint for the last one, what can you say about $(1-a)(1-b)$