Proofs using law of contraposition

Question

To prove a theorem my book says it does this:

"To prove a certain sentence "p" assume "not p", from this "q" is derived, then we show that "not q" holds, and therefore we accept the sentence "p" as true.

The theorem to prove is this: x+y=x+z → y=z

Which therefore represents the sentence "p" (which is of the form A→B), the thing I dont understand, is that at the beginning of the proof it says:

"Suppose the theorem were false, that is, x+y=x+z and yet y≠z"

but if "p" is the theorem, and it's an implication, ~p should not be A→~B it should be ~(A→B), what am I missing??

Thank you!

Answer 1

Well, the contraposition of $P\Rightarrow Q$ is $\neg Q\Rightarrow \neg P$.

In your case, $x+y=x+z \Rightarrow y=z$ has the contraposition $y\ne z\Rightarrow x+y\ne x+z$.

Answer 2

You are right in your statement that they have to show ~(A→B) and that is exactly what they have.

Note that the negation of $(A→B)$ is ($A$ and ~B) and that is exactly what they have.