TITLE: What's the difference between a contrapositive statement and a contradiction?
QUESTION [9 upvotes]: I keep mixing them up, because they are very similar.
Some contrapositives resemble some contradictions.

REPLY [6 votes]: When one speaks of a contrapositive or proving a contrapositive, one is speaking about the contrapositive of an implication (an "if...then" statement), and as pointed out in the earlier answers, if one wants to prove that $$P \implies Q\tag{1}$$ one can choose, instead, to prove $$\lnot Q \implies \lnot P,\tag{2}$$ because both statements are equivalent (i.e., if one is true, so is the other...and if one is false, so is the other). Don't confuse the appearance of the $\lnot$ symbol on each side of (2) as being either a negation of (1) nor contradiction. To see what I mean, one can correctly state that (1) (which does not contain the "$\lnot$" symbol) is the contrapostive of (2) because (2) is equivalent to $$\lnot(\lnot P) \implies \lnot(\lnot Q) \equiv P \implies Q.\tag{3}$$ 
In contrast, a contradiction is obtained when one derives or asserts that both a statement $P$ and its negation $\lnot P\;$ hold, i.e., when one asserts or derives: $$P \land \lnot P\tag{4}$$ (E.g., $x \in A \land x \notin A$ is a contradiction, and as such, is false regardless of whether or not $x \in A$). 
Another way of putting it is that a contradiction is any statement which is always false (i.e., a statement which is "inherently" false), and a contradiction can be thought of as the "opposite" of a tautology which is always true: e.g. $P \lor \lnot P$ is a tautology, and as such is true without knowing whether $P$ is true or false).