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We know that sometimes it is quite difficult to prove a mathematical statement; but it's contra-positive statement turns out to be easier. I am curious, why does it happen? Is there anything deep happening here?

Asaf Karagila
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abi
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    Sometimes is easier because we can go from $\forall$ to $\exists$. And proving that $\exists$ something is easier than (sometimes, not always) proving that something $\forall$. – Jika Jun 02 '14 at 15:31
  • Sometimes it's easier to prove that the negation of P implies the negation of Q, than to prove that Q implies P. – Doug Spoonwood Jun 02 '14 at 16:10

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Just as it is sometimes easier to prove a statement using a proof by contradiction, there are situations when proving the contrapositive of a statement falls out much more nicely than the statement itself.

For example, suppose we have a statement of the form $\forall x Px \implies \forall x Qx$.

The contrapositive is $$\lnot \forall x Qx \implies \lnot \forall x P x \equiv \exists x \lnot Q x \implies \exists x \lnot P x$$

In such cases, we need only prove existence of something that holds (or fails to hold) for some (we only need one) member in the domain, rather than having to prove something holds for all members in a domain.

EDIT: See also this post: When to use the contrapositive to prove a statement.

Community
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amWhy
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  • I don't see why you'd need the contra-positive here. You could just use the double negation law EpNNp to transform ∀xPx⟹∀xQx into $\lnot$$\lnot$∀xPx⟹$\lnot$$\lnot$∀xQx, and then use the quantifier exchange equivalences to get $\lnot$∃x $\lnot$Px⟹$\lnot$∃x$\lnot$Qx. Then again, proving the second form is probably harder than what you've referred to. – Doug Spoonwood Jun 02 '14 at 16:01
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I don't think there's anything deep happening here. As such, it's just another proof strategy.

The only explanation I can think of is that, de facto, in logic we are more interested in proving claims such as the implication from a universal statement to another universal statement, rather than statements involving existential quantification. And the former are easier to prove by contrapositive, as amWhy pointed out.

This fact might not be a mere contingency, but rather be a consequence of the fact that logic is universal, formal and topic-neutral (or this is what is usually thought). Thus, we happen to encounter many situations in which using proof by contrapositive is very useful, although this does not seem to be related with any intrinsic property of the method itself.

Cassidy
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