Absorption and Simplification
Absorption
This seemingly obscure relation turns out to be very useful in many proofs.
$a \wedge (a \vee b) = a$
\[\begin{align} &a \wedge (a \vee b) \\ &= (a \vee \color{red}F) \wedge (a \vee b) &&\text{Identity} \\ &= a \vee (\color{red}F \wedge b) &&\text{Distributivity} \\ &= a \vee \color{red}F &&\text{Base} \\ &= a &&\text{Identity} \end{align}\]$a \vee (a \wedge b) = a$
\[\begin{align} &a \vee (a \wedge b) \\ &= (a \wedge \color{green}T) \vee(a \wedge b) &&\text{Identity} \\ &= a \wedge (\color{green}T \vee b) &&\text{Distributivity} \\ &= a \wedge \color{green}T &&\text{Base} \\ &= a &&\text{Identity} \end{align}\]Simplification
This is less common than Absorption, but is still useful in some proofs
$a \wedge (\overline{a} \vee b) = a \wedge b$
\[\begin{align} &a \wedge (\overline{a} \vee b) \\ &= (a \wedge \overline{a}) \vee (a \wedge b) &&\text{Distributivity} \\ &= \color{red}F \vee (a \wedge b) &&\text{Noncontradiction} \\ &= a \wedge b &&\text{Identity} \end{align}\]$a \vee (\overline{a} \wedge b) = a \vee b$
\[\begin{align} &a \vee (\overline{a} \wedge b) \\ &= (a \vee \overline{a}) \wedge (a \vee b) &&\text{Distributivity} \\ &= \color{green}T \wedge (a \vee b) &&\text{Excluded Middle} \\ &= a \vee b &&\text{Identity} \end{align}\]