Denial of Universality/Examples/x less than or equal to 3/Examples/2, 3, 4

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

Let $P$ be the statement:

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

and $\lnot P$ its negation:

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

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

Then we have that:

$P$ is false

and consequently:

$\lnot P$ is true

Proof

The falsehood of $P$ can be demonstrated by citing $x \in S: x = 4$.

Thus $4$ is a counterexample to the assertion that all $x \in S$ are such that $x \le 3$.

Hence its negation is true.

$\blacksquare$

Sources

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