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

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Examples of Denial of Existence

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

Let $P$ be the statement:

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

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

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


Proof

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

$\blacksquare$


Examples

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

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


Example where $S = \closedint 0 3$

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


Sources