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

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

### 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