# Negation of Propositional Function in Two Variables

Jump to navigation
Jump to search

## 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:

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

\(\, \displaystyle \neg \forall x: \, \) | \(\displaystyle \exists y\) | \(:\) | \(\displaystyle \map P {x, y}\) | ||||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\, \displaystyle \exists x: \, \) | \(\displaystyle \neg \exists y\) | \(:\) | \(\displaystyle \map P {x, y}\) | Denial of Universality | ||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\, \displaystyle \exists x: \, \) | \(\displaystyle \forall y\) | \(:\) | \(\displaystyle \neg \map P {x, y}\) | 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