Alec C. answered • 03/22/21

Senior undergraduate - tutoring in math, CS, and English!

Hi, here's how I did it:

# ~(p **→** ~q) ∨ ( q ∧ ( p ∨ ¬ q))

This is the statement you've started with. We can build a truth table to show that ~(p → ~q) ≡ (p ∧ q) (try this yourself!). This gives us

# (p ∧ q) ∨ ( q ∧ ( p ∨ ¬ q))

Now, by the associative law, we can regroup the parentheses to get

# p ∧ (q ∨ ( q ∧ ( p ∨ ¬ q)))

Notice here that in the parentheses we have q ∨ (something). It doesn't really matter what that something evaluates to, since logically q or something = q. We thus have

# p ∧ (q)

which is what we needed to show.

Hope that helps! Let me know if you have any questions.

Alec C.

At that step I regrouped the parentheses, which is just the associative law. Nothing else at that step. I'm sure it's possible to solve this by working on the right side, but I think the way I did it above is probably the fastest/simplest way to do this problem.03/23/21

Alex V.

03/24/21

Stephanie W.

p ∧ (q ∨ ( q ∧ ( p ∨ ¬ q))) What law is used here? associative and ? Also is there a different way to do it?03/23/21