Contrapositive gives more info than implication?

Question

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Please forgive my childish drawing, this is the quickest way I could think to express my question.

Though the truth values of an implication and its contrapositive are the same, they do not seem to give the same amount of information, as represented in my horrible little sketch by which areas of the diagram are colour defined by each relation.

Isn't this a problem in their "equivalence" if one tells you more than the other? (It seems that the contrapositive describes a smaller set on the diagram, which I am equating to more information).

"If we're inside $p$, then we're inside $a$" carries exactly the same information as "If we're outside $a$ then we're outside $p$". — Arthur, Jan 17 '14 at 18:58

score 1 · Accepted Answer · answered Jan 17 '14 at 18:50

The reason you think that is that you don't draw $\lnot p$ in your diagram. If you think about sets, $p\rightarrow q$ means that $p\subseteq q$. The contrapositive means that the $q^c\subseteq p^c$, the complements have the reverse relation between them.

Contrapositive gives more info than implication?

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