Denial of Existence/Examples/x less than or equal to 3/Examples/0 to 3

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

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

Proof

The truth of $P$ can be demonstrated by citing $x \in S: x = 2$.

Thus $2$ is an instance of an $x \in S$ such that $x > 3$ that has been shown to exist.

Hence its negation is false.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $1 \ \text {(ii)} \, \text {(b)}$