Find cardinality for family of set For an arbitrary family of sets $A$ of two elemented subsets of $\mathbb{R}$ true is that: a) if $\bigcup A = \mathbb{R}$ then $|A|=|\mathbb{R}|$ b) if $|A|=|\mathbb{R}|$ then $\bigcap A = \emptyset$ I have problems with this ones how to approach them ?

HINT: For the first one it’s clear that if $\bigcup A=\Bbb R$, then $A$ is infinite, so $|A|+|A|=|A|$. And clearly $\left|\bigcup A\right|\le|A|+|A|$, so ... ?

The second one is false, so you should try to find a counterexample; there’s really only one sensible way to proceed once you have the idea of finding one.