Existential Statement

Existential Statement

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$$\exists x\in D,Q(x)$$ $$\exists x(x\in D\wedge Q(x))$$

• Made of Predicate Symbol, Predicate Variable & Quantifier
• true iff Q(x) is true for at least one x in D
• false iff Q(x) is false for all x in D
• For simple ones, usually existential comes with Logical Connectives

Important

$\exists x\in D,Q(x)$ is same as $Q(x_1)\vee Q(x_2)\vee \dots \vee Q(x_n)$.

Negation of Existential

$$\neg (\exists x\in D,Q(x))\equiv \forall x\in D,\neg Q(x)$$

• Negation of existential is Logical Equivalence to Universal Statement
• ALL ARE NOT

Unique Existence

$$\exists !x\in D,P(x)$$

• There exists an unique (one and only one) $x$, such that $P(x)$ is true