Laws of Inference question?

Question

Is the following "true" or "false"

(p -> q) ^ p

Oh, and can you explain why!?!?

THIS STUFF IS DRIVING ME NUTS! :(
I'm like the only person in Geometry class to not understand this T_T



thanks so much ; ;!

Answer 1

(p -> q) is equivalent to ~p v q, where ~ means "not". Therefore, it's true if p is false or q is true.

It follows that the original statement, (p->q) ^ p, is true if p is true and (p is false or q is true), meaning q must be true as well.

Effectively, this statement is logically equivalent to p ^ q, because both statements have the same truth value.

Another way to see it is this: write the statement as

(~p v q) ^ p

and use the distributive law to obtain

((~p ^ p) v (q ^ p))

but ~p ^ p is false, and (x v false) is equivalent to x, so you're left with p v q.

As for why it's driving you nuts, it doesn't have to. You work with this for enough problems and you can get it to click, but it's going to take a lot of patience, and diligence. But you can do it.

Answer 2

It is true only when p and q are both true.
I used a truth table to find the truth value. If you don't know what a truth table is, then look at it this way:
An "and" statement is only true when both parts are true.
So (p->q) is true and p is true.
p->q is only false when p is true and q is false. Otherwise, p->q is true. From that you get that q must be true.

Answer 3

Us answering won't help you