How to simplify this Boolean expression?

Question

The expression not p or p and q can be simplified to not p or q. Please show me the steps and rules which are used for this simplification.

Answer 1

$\neg p \lor (p \land q) \Leftrightarrow$ (Distribution)

$(\neg p \lor p) \land (\neg p \lor q) \Leftrightarrow$ (Complement)

$\top \land (\neg p \lor q) \Leftrightarrow$ (Identity)

$\neg p \lor q$

This is actually such a common pattern that there is a 1 step equivalence principle for it:

Reduction

$\neg p \lor (p \land q) \Leftrightarrow \neg p \lor q$

and its dual:

$\neg p \land (p \lor q) \Leftrightarrow \neg p \land q$

Answer 2

(1) In standard syntaxes, "not p or p and q" is ambiguous between "(not p or p) and q" and "not p or (p and q)". You need to be clear which is meant -- presumably in this case the second.

(2) You presumably know the distributive law that tells you that A or (B and C) is equivalent to (A or B) and (A or C). Ask yourself what happens if you apply the law in the present case?

(3) In particular, when you apply the law, what can you do to the first conjunct (corresponding to (A or B))?