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