Denial of Existence/Examples/x less than or equal to 3/Examples

Examples of Denial of Existence: $\forall x \in S: x \le 3$

Example where $S = \set {2, 3, 4}$

Let $P$ be the statement:

$\exists x \in S: x \le 3$

and $\lnot P$ its negation:

$\forall x \in S: x > 3$

Let $S = \set {2, 3, 4}$.

Then we have that:

$P$ is true

and consequently:

$\lnot P$ is false

Example where $S = \closedint 0 3$

Let $P$ be the statement:

$\exists x \in S: x \le 3$

and $\lnot P$ its negation:

$\forall x \in S: x > 3$

Let $S = \closedint 0 3$ where $\closedint \cdot \cdot$ denotes a closed real interval.

Then we have that:

$P$ is true

and consequently:

$\lnot P$ is false