Logical Equivalence: prove that P OR (Q AND P) <=> P

[pip1]

How can I prove that P OR (Q AND P) <=> P
without using truth tables?

 

[pka]

This can be done by the technique known as ‘Indirect Proof’.
Assume not P and get a contradiction.

 

[pip1]

So ~P <=> ~[P ^ (P v Q)]

 

[pka]

\(\displaystyle \begin{array}{l}
P \vee \left( {P \wedge Q} \right)\quad \mbox{given} \\
\neg P\quad \mbox{indirect proof}\\
P \wedge Q\quad \mbox{disjunction} \\
P\quad \mbox{simplification} \\
P \wedge \neg P\quad \mbox{conjunction} \\
P\quad \mbox{contradiction} \\
\end{array}\)

\(\displaystyle \begin{array}{l}
P \quad \mbox{given}\\
P \vee \left( {P \wedge Q} \right) \quad \mbox{disjunction}\\
\end{array}\)