Negation of Propositional Function in Two Variables

Theorem

Let $\map P {x, y}$ be a propositional function of two Variables.

Then:

$\neg \forall x: \exists y: \map P {x, y} \iff \exists x: \forall y: \neg \map P {x, y}$

That is:

It is not the case that for all $x$ a value of $y$ can be found to satisfy $\map P {x, y}$

means the same thing as:

There exists at least one value of $x$ such that for all $y$ it is not possible to satisfy $\map P {x, y}$

Proof

\(\ds \neg \forall x: \, \)

\(\ds \exists y\)

\(:\)

\(\ds \map P {x, y}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists x: \, \)

\(\ds \neg \exists y\)

\(:\)

\(\ds \map P {x, y}\)

Denial of Universality

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists x: \, \)

\(\ds \forall y\)

\(:\)

\(\ds \neg \map P {x, y}\)

Denial of Existence

$\blacksquare$

Sources

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers