Boolean Logic - Reduction - $a \vee (a \wedge b) = a$

Question

How would I simplify / reduce the following equation using boolean identities/proofs? $$a \vee (a \wedge b) = a$$ So far I've used the distributivity identity and got $$(a\vee a) \wedge (a\vee b)$$ I then used the idempotence law to get $$a \wedge (a\vee b)$$ Now I can't think of what to do next.

Answer 1

You're probably looking for this and not for truth-tables or what I'd written previously:

$$a \lor (a \land b)$$ $$(a \land \top) \lor (a \land b)$$ $$a \land (\top \lor b)$$ $$a \land \top$$ $$a$$

Answer 2

$$ \begin{array}{ccc} a & b & a\mathrm{~AND~}b & a\mathrm{~OR~}(a\mathrm{~AND~}b)\\\hline 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 1\\ 1 & 1 & 1 & 1\\ \end{array} $$