(P and(not(not P or Q))) or( P and Q) equals P

Question

I've been trying to verify the condition above but I get stuck on the passage :

$$(P \land (P \land \lnot Q)) \lor (P \land Q)$$

I don't know how to simplify it since there are two ands and a not Q.

I thought I might use absorption law, but it states that the signs should be different ( in other words, it should be and- or and viceversa). What do you think?

Answer 1

Hint: The next step is(/steps are) $a\wedge (a\wedge b) = (a\wedge a)\wedge b = a\wedge b$.   (Association and Idempotence)

Answer 2

(P∧(P∧¬Q))∨(P∧Q)

First use the associative law on (P∧(P∧¬Q)) to get ((P∧P)∧¬Q)). Which law do you apply then to get (P∧¬Q) for the first part?

Then we have (P∧¬Q)∨(P∧Q). Use one of the distributive laws to condense this.

Do you see how to proceed from there?