# Denial of Universality/Examples/x less than or equal to 3

## Example of Denial of Universality

Let $S \subseteq \R$ be a subset of the real numbers.

Let $P$ be the statement:

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

The negation of $P$ is the statement written in its simplest form as:

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

## Proof

 $\displaystyle$  $\displaystyle \lnot \forall x \in S: x \le 3$ $\displaystyle$ $\leadsto$ $\displaystyle \exists x \in S: \lnot x \le 3$ Denial of Universality $\displaystyle$ $\leadsto$ $\displaystyle \exists x \in S: x \not \le 3$ $\displaystyle$ $\leadsto$ $\displaystyle \exists x \in S: x > 3$

$\blacksquare$

## Examples

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

Let $P$ be the statement:

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

and $\lnot P$ its negation:

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

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

Then we have that:

$P$ is false

and consequently:

$\lnot P$ is true

### Example where $S = \closedint 0 3$

Let $P$ be the statement:

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

and $\lnot P$ its negation:

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

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