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

Example of Denial of Universality

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

Let $P$ be the statement:

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

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

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

Proof

\(\ds \)

\(\)

\(\ds \lnot \forall x \in S: x \le 3\)

\(\ds \)

\(\leadsto\)

\(\ds \exists x \in S: \lnot \paren {x \le 3}\)

Denial of Universality

\(\ds \)

\(\leadsto\)

\(\ds \exists x \in S: x \not \le 3\)

\(\ds \)

\(\leadsto\)

\(\ds \exists x \in S: x > 3\)

$\blacksquare$

Examples

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

Sources

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