The contrapositive of the implication 
is the implication
. These are equivalent forms. Sometimes it
is easier to prove a result in the contrapositive form rather than in the
original conditional form. Proving the contrapositive might also be done
as a direct proof, by induction or by contradiction. The only care that
one need take is that the contrapositive is formed correctly. For example
the contrapositive of the statement ``If x is odd then 5x is odd'' is
the statement ``If 5x is not odd then x is not odd'' or better still,
``If 5x is even then x is even''. Some statements become more difficult
to put in the contrapositive form when they involve several connectives.
The contrapositive of the statement
``If x is odd and y is even then xy is even''
is
``If xy is odd then x is even or y is odd''.
From our table of equivalents in section 2, we see this is equivalent to
``If xy is odd and x is odd then y is odd''.
Let's realize that in a direct proof of this result, our assumed true statement would change depending on which form we chose. For the statement ``If x is odd and y is even then xy is even'' we would assume that x is and odd integer and y an even integer. For the statement ``If xy is odd then x is even or y is odd'' we would assume that the product xy is an odd integer. Finally, for the statement ``If xy is odd and x is odd then y is odd'' we would assume that xy is odd and x is odd.
Whether it is better to prove a statement in the original form or to rewrite it to an equivalent form is difficult to answer. Certainly when answering problems from a text, it would be prudent to keep to the original form of the statement in the majority of cases. However, if you get stuck on a proof it might be beneficial to consider alternative phrasings.