which of the following subsets of $\mathbb R^2$ are uncountable. 
*

*$\{\,(a,b)\in \mathbb R^2\mid a\leq b\,\}$

*$\{\,(a,b)\in\mathbb R^2\mid a+b\in\mathbb Q\,\}$

*$\{\,(a,b)\in \mathbb R^2\mid ab\in \mathbb Z\,\}$

*$\{\,(a,b)\in\mathbb R^2\mid a,b\in \mathbb Q\,\}$.


I know option 1 is uncountable and option 4 is countable. 
I think option 2, 3 are uncountable but I am not sure. Can someone help me?

$\mathbb{Q}$ being countable,$\mathbb{Q^2}$ is also countable and consequently $4$ is countable.Choosing a=y,b=n/y where n is a natural number and y is a positive irrational and less than 1 we can satisfy the requirement of $3$ in uncountably many ways, irrationals in ($0,1$) being uncountable.
$a=x-y,b=x+y \text{ (where x is rational and y irrational) }\Rightarrow a+b=2x\in \mathbb{Q}$ i.e there are uncountably many choices for 2. Choosing a and b both irrationals we can satisfy the requirement of 1 in uncountably many ways.So $1,2,3$ are uncountable. 
