Denial of Existence/Examples/x less than or equal to 3/Examples/2, 3, 4
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Example of Denial of Existence: $\forall x \in S: x \le 3$
Let $P$ be the statement:
- $\exists x \in S: x \le 3$
and $\lnot P$ its negation:
- $\forall x \in S: x > 3$
Let $S = \set {2, 3, 4}$.
Then we have that:
- $P$ is true
and consequently:
- $\lnot P$ is false
Proof
The falsehood of $\lnot P$ can be demonstrated by citing $x \in S: x = 2$.
Thus $4$ is a counterexample to the assertion that all $x \in S$ are such that $x > 3$.
Hence its negation $P$ is true.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $1 \ \text {(ii)} \, \text {(a)}$