TITLE: How do I prove $A \cup\varnothing = A$ and $A \cap\varnothing = \varnothing$
QUESTION [5 upvotes]: $$A \cup \emptyset = A \qquad \text{and} \qquad A \cap \emptyset = \emptyset$$
I know these are identities but I'm not sure how to get started.

REPLY [5 votes]: These tautologies are useful. 
$$P\wedge T \iff P$$
and 
$$P\vee F \iff P.$$