Mathematical Logic (Tautology)?

Question

Just got my first book on mathematical logic. I am learning about the tautologies. I am a little confused as to what the law of contraposition (p->q)<->(~q->~p) shows. I noticed that:
(p<->q)<->(~p<->~q)
(p<->q)<->(~q<->~p)
(p->q)->(~q->~p)
also seem to be similar tautologies, so what makes the Law of Contraposition in the form it was given special?
The book explains that the law of contradiction means that p and ~p cannot occur simultaniously and then goes on to the Law of the Excluded Middle, but it does not seem to give such attention to Contraposition. Any help or links would be appreciated.

Answer 1

I agree with the previous answerer...I think that the Law of Contraposition is, in some sense, stronger than the other ones you've listed, because the tautologies you've listed can easily be proved using the Law of Contraposition.

The Law of Contraposition means that, if I wanted to prove that, say, "all cows eat grass," then if I wanted to, I could instead show "If something does not eat grass, then it is not a cow." Both statements say essentially the same thing: That the class of all cows is entirely contained within the class of all things that eat grass.

Contraposition can be very useful in writing proofs. Suppose I want to prove the statement "If p(x) is a polynomial with infinitely many roots, then it must be that p(x) = 0." I can't really prove this directly (I'd have to start with some arbitrary polynomial with infinitely many roots, and show it was equal to 0--it is not really clear how to do this), so instead I would show that if p(x) isn't the zero polynomial, then it only has finitely many roots; then by contraposition I know my original statement was true.

Answer 2

(p->q)<->(~q->~p) actually implies the three other tautologies you list, but is more general than any of them. That's why this form is used.

Contraposition is a little harder to explain in words, but it's essentially about reversing the information flow. You could think of it as saying "If A always implies B, then if B didn't happen A cannot have happened." In sets, it's saying that A ⊆ B <=> Bc ⊆ Ac where c indicates the complement.

An example is the statement "If it is Monday, the computer will break." The contrapositive of this is the statement "If the computer does not break, it is not Monday." These two statements are logically equivalent.