Negated Restricted Universal Quantifier
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Theorem
Let $x$ and $A$ be sets.
Let $\map P x$ be a propositional function.
- $\neg \forall x \in A : \map P x \iff \exists x \in A : \neg \map P x $
Proof
Sufficient Condition
| \(\ds \neg \forall x \in A: \, \) | \(\ds \) | \(\map P x\) | \(\ds \) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \neg \forall x: \, \) | \(\ds \) | \(x \in A \implies \map P x\) | \(\ds \) | Definition of Restricted Universal Quantifier | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exists x: \, \) | \(\ds \) | \(\neg \paren{x \in A \implies \map P x}\) | \(\ds \) | De Morgan's Laws (Predicate Logic)/Denial of Universality | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exists x: \, \) | \(\ds \) | \(x \in A \land \neg \map P x\) | \(\ds \) | Conjunction with Negative is Equivalent to Negation of Conditional | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exists x \in A: \, \) | \(\ds \) | \(\neg \map P x\) | \(\ds \) | Definition of Restricted Existential Quantifier |
$\Box$
Necessary Condition
| \(\ds \exists x \in A: \, \) | \(\ds \) | \(\neg \map P x\) | \(\ds \) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exists x: \, \) | \(\ds \) | \(x \in A \land \neg \map P x\) | \(\ds \) | Definition of Restricted Existential Quantifier | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exists x: \, \) | \(\ds \) | \(\neg \paren{x \in A \to \map P x}\) | \(\ds \) | Conjunction with Negative is Equivalent to Negation of Conditional | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \neg \forall x: \, \) | \(\ds \) | \(x \in A \to \map P x\) | \(\ds \) | De Morgan's Laws (Predicate Logic)/Denial of Universality | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \neg \forall x \in A: \, \) | \(\ds \) | \(\map P x\) | \(\ds \) | Definition of Restricted Universal Quantifier |
$\blacksquare$