Predicates and quantifiers

Quantifiers: 
∀x P(x) ≡ P(x1) ∧ P(x1) ∧ P(x1) ∧ P(x1) ∧.....P(xn)
∃x P(x) ≡ P(x1) ∨ P(x1) ∨ P(x1) ∨ P(x1) ∨.....P(xn)

Statement When True ? When False ?
∀x P(x) P(x) is true for every x There is an x for which P(x) is false 
∃x P(x) There is an x for which P(x) is true. P(x) is false for every x

Uniqueness quantifier

∃! x P(x) ≡ There exists a unique x such that P(x) is true.
De Morgan's Laws for Quantifiers:
¬∃x P(x) = ∀x ¬P(x)
¬∀x P(x) = ∃x ¬P(x)
Quantifications of two variables:

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