Nested Quantifiers

Yo dawg I heard you like quantifiers

So I put some quantifiers in your quantifiers

∀x ∀y ∃z x * y = z 

(where the domain of x, y, and z are the real numbers)

Let's break it down

∀x ∀y ∃z x * y = z

  • ∀x P(x) where P(x) = "∀y ∃z x * y = z"
  • ∀x ∀y Q(y) where Q(y) = "∃z x * y = z"
  • ∀x ∀y ∃z R(z) where R(z) = "x * y = z"

∀x ∀y ∃z x * y = z

  • What if it were "∀z"? 
  • What if we did "∃z ∀x ∀y"?

Two Variable Quantifiers

Let P(x, y) be "x has seen movie y" where x are students in this class and y are movies on Netflix.

  • ∀x ∀y P(x, y)
  • ∀x ∃y P(x, y)
  • ∃x ∀y P(x, y)
  • ∃x y P(x, y)

Let's Get Crazier

Let P(x) = "x owns a PS4" and Q(x, y) = "x played y"

x is students in this class, y is PS4 games


  • ∃x P(x) ^ ¬Q(x, Uncharted 4)
  • ∀x P(x) → Q(x, y)
  • ∃x P(x) ^ ¬Q(x, y)
  • ∃x P(x) ^ ¬Q(x, y)

How do we express "Exactly two students own PS4s and played Destiny"?

The Uniqueness Quantifier

  • Recall "∃! x P(x)" means there exists exactly 1 x
  • We can express this in terms of our other quantifiers
  • ∃! x P(x) ↔ ∃ x P(x) ^  y (P(y)  y = x)

Exactly two students own PS4s and played Destiny

  1. ∃ x P(x) ^ Q(x, Destiny)
  2. ∃ x ∃ y (P(x) ^ Q(x, Destiny)) ^ (P(y) ^ (Q(y, Destiny))
  3. ∃ x ∃ y (P(x) ^ Q(x, Destiny)) ^ (P(y) ^ (Q(y, Destiny))
    ∀ z (P(z) ^ Q(z, Destiny)) → (z = x v z = y)

Negating Nested Quantifiers

DeMorgan's Laws all the way down

  1. ¬∃x ∃y ∀z P(x, y, z)
    think of it as ¬∃x P(x)
  2. ∀x ¬∃y ∀z P(x, y, z)
    ∀x ¬P(x)
  3. ∀x ∀y ¬∀z P(x, y, z)
  4. ∀x ∀y ∃z ¬P(x, y, z)

Nested Quantifiers

By George Lee

Nested Quantifiers

I heard you like quantifiers

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