Logical Proof Question!!?

Question

X -> N
~X -> P
( M ^ N) -> Q
~Q
M
_____________
therefore P

I'm not even sure where to start :( Any help would be greatly appreciated!!!!!

Answer 1

First of all, you have conditions of M and ~Q (not Q). According to another condition (M^N)->Q (M AND N implies Q). However you have the condition of ~Q. Therfore, only M or N exits but you're given that M exits. hence N does not exits. According to another condition X->N but N does not exits. hence we have ~X (not X condition). Well, since ~X -> P (not X implies P). Therefore, the answer is P!

In turns of mathematical symbols:

(~Q ^ M) -> ~N {because ~Q, M, and (M^N)->Q}
~N ->~ X {because X->N}
Therefore P {because ~X->P}

Answer 2

First remember that A -> B is equivalent to ~A v B, as shown below:

A . B . ~A . (A -> B) . (~A v B)
F . F ... T ........ T .......... T
F . T ... T ........ T .......... T
T . F ... F ........ F .......... F
T . T ... F ........ T .......... T

So convert your statements as follows:

~X v N
X v P
~(M ^ N) v Q
~Q
M

Also ~(A ^ B) is equivalent to ~A v ~B, use that for #3

~X v N
X v P
~M v ~N v Q
~Q
M

Now look at #3. ~M is false (because M is true in #5), Q is also false (because ~Q is true in #4). That requires ~N to be true.

Now look at #1, N is false (because ~N is true from above). That means ~X must be true.

Finally look at #2. X is false (because ~X is true as shown above). That means that P must be true.

Hence P is true, as was to be proven.

By the way, it's been a few years since I had to do logical proofs, and I know I've helped you on a few recently. I wanted to say thank you for letting me "dust off the cobwebs" and have fun with them again. (I know that sounds crazy, but they are like little puzzles... which my brain finds "fun"). Anyway, if you have more, I'll gladly help.

Answer 3

Oddly enough, I have never done problems like this before. So, please forgive my assumptions.

First off, you have M. You also know (M ^ N) -> Q.
But you have ~Q. Since M is true, then N is not the true statement. Or, more specifically ~(M ^ N) -> ~M v ~N. You have M, so you have ~N.

We know that X -> N is also ~N -> ~X. Since have have ~N, we have ~X. And from the given, we have ~X -> P.

Answer 4

The implication (M ^ N) -> Q is equivalent to (by contraposition)
~Q -> ~(M ^ N)

Since you are given ~Q, you can conclude
~(M ^ N)

which is equivalent to
M -> ~N

Since you are told that M is true, you can conclude
~N

The implication X -> N is equivalent to ~N -> ~X. It follows that
~X

and using ~X -> P, we finally conclude P.

Answer 5

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