There Exists

A Quantifier. $\exists$ There exists atleast one. It is applied immediately to the free variable after, $\exists x$, rather than for the entire statement similar to how $\neg$ is only for a single term after

Existentially Quantified

$\exists x,P(x)$

• True if P(x) is true for some element in the Universe of Discourse
• False otherwise

Quantifier Negation Laws

$\neg \forall ,P(x)\equiv \exists x,\neg P(x)$ $\neg \exists ,P(x)\equiv \forall x,\neg P(X)$ Negate the predecates, and the quantifier

Negating Predicates with Negated Quantifiers

$\exists x\neg \forall y,C(y,x)$

1. Remove all negations in front of quantifiers
2. Flip quantifiers

Proof Writing

$\exists x\in D$ such that $D(x)$ Choose x = __ (this could be a variable or concrete value) then prove both $x\in D$ and $P(x)$ is true.

Unique Existence

• Unique Existence