Hey guys. I don't understand how to prove this:
(a)
$\displaystyle
\text A \cup (null) = A
$
(b)
$\displaystyle
\text A \cap U = A
$
one method: Let $\displaystyle x \in A \cup \emptyset$ ......we assume A is non-empty, it's trivial otherwise
$\displaystyle \Rightarrow x \in A$ or $\displaystyle x \in \emptyset$
we cannot have $\displaystyle x \in \emptyset$, thus, $\displaystyle x \in A$
therefore, $\displaystyle (A \cup \emptyset ) \subset A$
now assume $\displaystyle x \in A$. this implies $\displaystyle x \in A \cup \emptyset$ (i'm sure there's some kind of axiom or something that allows us to say this). thus $\displaystyle A \subset (A \cup \emptyset )$
since $\displaystyle (A \cup \emptyset ) \subset A$ and $\displaystyle A \subset (A \cup \emptyset )$, we have $\displaystyle A \cup \emptyset = A$