Converse and Contrapositive Statements/Examples

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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
$P$ and $R$ are true
$Q$ and $S$ are false.

Triangles of 2 and 3 Equal Sides

Let $P$ be the statement:

If a triangle has $3$ equal sides, then it has $2$ equal sides

which is trivially true.

$P$ can also be written:

A triangle has $3$ equal sides only if it has $2$ equal sides
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:

If a triangle has $2$ equal sides, then it has $3$ equal sides

which is false.