# Converse and Contrapositive Statements/Examples

## Examples of Converse Statements and Contrapositive Statements

### $x < 5$ and $x \le 5$

Let:

- $P$ be the statement $x < 5 \implies x \le 5$
- $Q$ be the statement $x \le 5 \implies x < 5$
- $R$ be the statement $x > 5 \implies x \ge 5$
- $S$ be the statement $x \ge 5 \implies x > 5$

for $x \in \R$.

Then:

- $P$ and $Q$ are converse statements
- $R$ and $S$ are converse statements

- $P$ and $R$ are contrapositive statements
- $Q$ and $S$ are contrapositive statements

### Triangles of 2 and 3 Equal Sides

Let $P$ be the statement:

which is trivially true.

$P$ can also be written:

*A***sufficient condition**for a triangle to have $2$ equal sides is for it to have $3$ equal sides

*A***necessary condition**for a triangle to have $3$ equal sides is for it to have $2$ equal sides

The converse of $P$ is:

which is false.