Proof_by_Contrapositive
Definition
From Wikipedia:
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contrapositive. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
The contrapositive of $P \rightarrow Q$ is $\neg Q \rightarrow \neg P$.
Intuitive Example
In the Euler diagram shown, if something is in $A$, it must be in $B$ as well. So we can interpret “all of $A$ is in $B$” as: $A \rightarrow B$.
It is also clear that anything that is not within $B$ cannot be within $A$, either. This statement, which can be expressed as: $\neg B \rightarrow \neg A$ is the contrapositive of the above statement. Therefore, one can say that $\left. (A \rightarrow B) \right. \leftrightarrow \left. (\neg B \rightarrow \neg A) \right.$.
