"Write a Logical Proof for this Argument"?

Question

Has anyone learned about logical proofs before? It looked pretty simple on the blackboard, but now that I'm actually trying it, its MAD CONFUSING!!!!

Here's the question

A -> B
~(C^B)
C
__________
~A

Can someone tell me where to start or at give me a website about it? THANKS!!!

Answer 1

~ means "not"
^ means "and"
v means "or"
-> means "then" or "implies"

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

Also ~(C ^ B) is equivalent to ~C v ~B (by DeMorgan's law). You can prove this by creating a table as shown above, or just use it. When you distribute the ~ through, remember to flip the ^ (and) to v (or).

So you have the following equivalent set of statements:
~A v B
~C v ~B
C

It's always easiest to start with the line that only has one value, namely #3. This means that C is true.

Now look at #2. Because C is true, ~C is false, therefore ~B must be true to make the statement true.

Finally look at #1. B is false (because ~B was shown true above). Again ~A must be true to make the statement true.

Thus ~A is true, as was to be proven.

Answer 2

A -> B is the same as ~B -> ~A Law of Contrapostive

~(C and B) is the same as ~C or ~B DeMorgan's Law

Since C is True, ~B Must be true and thus ~A must be true, since ~B -> ~A

Answer 3

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