# Thread: Problem with logic simplification

1. ## Problem with logic simplification

I have a problem that requires me to simplify some logic.
The problem is (¬p ∧ q) ∧ (¬q ∨ ¬r)

I thought about distributing everything but there are different signs (^ and v) in the individual parentheses so I'm not sure how to multiply those together. If it was (¬p ∧ q) ∧ (¬q ∧ ¬r) I would've been able to simplify it.

The answer is ¬p ∧ q ∧ ¬r

2. ## Re: Problem with logic simplification

Use distribution:
(¬p ∧ q) ∧ (¬q ∨ ¬r) <==> (¬p ^ q ^ ¬q) v (¬p ∧ q ∧ ¬r)

and then see where that leads you.

3. ## Re: Problem with logic simplification

You'll need those two q's to somehow combine, and the quickest way to "stick them together so they might combine" is to use associtivity of "and" to pull out the "not p" and mush the q-stuff together. After the first statement, it should be clear.

(¬p ∧ q) ∧ (¬q ∨ ¬r)

¬p ∧ [ q ∧ (¬q ∨ ¬r) ]

Spoiler:

¬p ∧ [ (q ∧ ¬q) ∨ (q ∧ ¬r) ]

¬p ∧ [ (F) ∨ (q ∧ ¬r) ]

¬p ∧ [ (q ∧ ¬r) ]

¬p ∧ q ∧ ¬r

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