Nested Quantifiers

What are Nested Quantifiers?

Nested quantifiers occur when one quantifier appears within the scope of another quantifier. They are essential for expressing complex mathematical and logical statements.

General Form: ∀x ∃y P(x, y) or ∃x ∀y Q(x, y), etc.

Understanding the Order of Quantifiers

The order of quantifiers is crucial and changes the meaning of a statement.

Example: The Importance of Order

Consider the predicate L(x, y): “x loves y”

1. ∀x ∃y L(x, y)

   • Meaning: “For every person x, there exists a person y such that x loves y”
   • In simple words: “Everyone loves someone (possibly different people)”
   • Example: Ram loves Sita, Mohan loves Gita (different people)

2. ∃y ∀x L(x, y)

   • Meaning: “There exists a person y such that for every person x, x loves y”
   • In simple words: “There is someone who is loved by everyone”
   • Example: Everyone loves the same person

These are NOT equivalent! The first is likely true, the second is very restrictive.

Reading Nested Quantifiers

Read nested quantifiers from left to right, respecting the scope.

Basic Pattern

∀x ∃y P(x, y) reads as: “For every x, there exists a y such that P(x, y)“

Step-by-Step Reading

1. Start with the leftmost quantifier
2. Read each quantifier with its variable
3. End with the predicate

Common Nested Quantifier Patterns

Pattern 1: ∀x ∀y P(x, y)

Meaning: “For all x and for all y, P(x, y) is true”

Example 1:

• Statement: ∀x ∀y (x + y = y + x)
• Meaning: “For all numbers x and y, x + y equals y + x”
• In words: “Addition is commutative”
• Truth Value: TRUE

Example 2:

• Statement: ∀x ∀y (x < y)
• Meaning: “For all x and y, x is less than y”
• Truth Value: FALSE (not every x is less than every y)

Pattern 2: ∀x ∃y P(x, y)

Meaning: “For every x, there exists a y such that P(x, y)”

Example 1:

• Statement: ∀x ∃y (x + y = 0)
• Meaning: “For every number x, there exists a number y such that x + y = 0”
• In words: “Every number has an additive inverse”
• Truth Value: TRUE (y = -x)

Example 2:

• Statement: ∀x ∃y (x < y)
• Meaning: “For every number x, there exists a number y such that x < y”
• In words: “For any number, there’s a larger number”
• Truth Value: TRUE (no largest number exists)

Pattern 3: ∃x ∀y P(x, y)

Meaning: “There exists an x such that for all y, P(x, y) is true”

Example 1:

• Statement: ∃x ∀y (x + y = y)
• Meaning: “There exists a number x such that for all y, x + y = y”
• In words: “There exists an additive identity”
• Truth Value: TRUE (x = 0)

Example 2:

• Statement: ∃x ∀y (x × y = y)
• Meaning: “There exists a number x such that for all y, x times y = y”
• In words: “There exists a multiplicative identity”
• Truth Value: TRUE (x = 1)

Example 3:

• Statement: ∃x ∀y (x < y)
• Meaning: “There exists a number x that is less than all numbers y”
• In words: “There is a smallest number”
• Truth Value: FALSE (for real numbers, no smallest exists)

Pattern 4: ∃x ∃y P(x, y)

Meaning: “There exists an x and there exists a y such that P(x, y)”

Example 1:

• Statement: ∃x ∃y (x + y = 10)
• Meaning: “There exist numbers x and y such that x + y = 10”
• Truth Value: TRUE (e.g., x = 3, y = 7)

Example 2:

• Statement: ∃x ∃y (x² + y² = 25)
• Meaning: “There exist numbers x and y such that x² + y² = 25”
• Truth Value: TRUE (e.g., x = 3, y = 4 or x = 0, y = 5)

Order Matters: Detailed Comparison

Example 1: Numbers

Let the domain be all real numbers.

Example 2: Relationships

Let F(x, y) = “x is a friend of y”

Negating Nested Quantifiers

To negate nested quantifiers, apply the negation rules repeatedly from left to right.

Rules

1. ¬∀ becomes ∃¬
2. ¬∃ becomes ∀¬

Example 1: Negating ∀x ∃y P(x, y)

Step 1: Apply negation to first quantifier ¬(∀x ∃y P(x, y)) ≡ ∃x ¬(∃y P(x, y))

Step 2: Apply negation to second quantifier ≡ ∃x ∀y ¬P(x, y)

Result: ∃x ∀y ¬P(x, y)

English Example:

• Original: “Everyone has someone they love”
• Negation: “There’s someone who doesn’t love anyone”

Example 2: Negating ∃x ∀y P(x, y)

Step 1: ¬(∃x ∀y P(x, y)) ≡ ∀x ¬(∀y P(x, y))

Step 2: ≡ ∀x ∃y ¬P(x, y)

Result: ∀x ∃y ¬P(x, y)

English Example:

• Original: “There’s someone who is loved by everyone”
• Negation: “For everyone, there’s someone who doesn’t love them”

Example 3: Complex Negation

Original: ∀x ∃y ∀z ((x < y) ∧ (y < z))

Step 1: ¬(∀x ∃y ∀z ((x < y) ∧ (y < z)))

Step 2: ≡ ∃x ¬(∃y ∀z ((x < y) ∧ (y < z)))

Step 3: ≡ ∃x ∀y ¬(∀z ((x < y) ∧ (y < z)))

Step 4: ≡ ∃x ∀y ∃z ¬((x < y) ∧ (y < z))

Step 5: Apply De Morgan’s Law ≡ ∃x ∀y ∃z ((x ≥ y) ∨ (y ≥ z))

Translating English to Nested Quantifiers

Example 1: Mathematics

English: “For every positive number, there exists a smaller positive number.”

• Let P(x) = “x is positive”
• Translation: ∀x (P(x) → ∃y (P(y) ∧ y < x))

Example 2: Database Query

English: “Every student has enrolled in at least one course.”

• Let S(x) = “x is a student”
• Let C(y) = “y is a course”
• Let E(x, y) = “x is enrolled in y”
• Translation: ∀x (S(x) → ∃y (C(y) ∧ E(x, y)))

Example 3: Social Network

English: “There is a person who knows everyone.”

• Let K(x, y) = “x knows y”
• Translation: ∃x ∀y K(x, y)

OR (if we exclude self-knowledge):

• Translation: ∃x ∀y (x ≠ y → K(x, y))

Example 4: Real Analysis

English: “For every ε > 0, there exists a δ > 0 such that if |x - a| < δ, then |f(x) - L| < ε.”

This is the definition of a limit!

• Translation: ∀ε (ε > 0 → ∃δ (δ > 0 ∧ ∀x (|x - a| < δ → |f(x) - L| < ε)))

Three or More Quantifiers

Example 1: Transitivity

Statement: ∀x ∀y ∀z ((x < y) ∧ (y < z) → (x < z))

Meaning: “For all x, y, and z, if x < y and y < z, then x < z”

In words: “Less than is transitive”

Example 2: Between

Statement: ∀x ∀z ((x < z) → ∃y ((x < y) ∧ (y < z)))

Meaning: “For any two different numbers, there’s a number between them”

In words: “The real numbers are dense”

Example 3: Calculus - Continuity

Statement: ∀ε (ε > 0 → ∃δ (δ > 0 ∧ ∀x (|x - c| < δ → |f(x) - f(c)| < ε)))

Meaning: This is the definition of continuity at point c.

Truth Values with Finite Domains

Example: Domain = {1, 2}

Let P(x, y) = “x + y = 3”

Evaluate: ∀x ∃y P(x, y)

Expansion:

• For x = 1: ∃y P(1, y) = P(1, 1) ∨ P(1, 2) = (1+1=3) ∨ (1+2=3) = F ∨ T = T
• For x = 2: ∃y P(2, y) = P(2, 1) ∨ P(2, 2) = (2+1=3) ∨ (2+2=3) = T ∨ F = T

Final: T ∧ T = TRUE

Evaluate: ∃x ∀y P(x, y)

Expansion:

• For x = 1: ∀y P(1, y) = P(1, 1) ∧ P(1, 2) = F ∧ T = F
• For x = 2: ∀y P(2, y) = P(2, 1) ∧ P(2, 2) = T ∧ F = F

Final: F ∨ F = FALSE

Quantifier Equivalences

Commuting Same Quantifiers

1. ∀x ∀y P(x, y) ≡ ∀y ∀x P(x, y)

   • Two universal quantifiers can be swapped

2. ∃x ∃y P(x, y) ≡ ∃y ∃x P(x, y)

   • Two existential quantifiers can be swapped

Cannot Commute Different Quantifiers

∀x ∃y P(x, y) NOT≡ ∃y ∀x P(x, y)

These mean different things (as we saw earlier)!

Practical Applications

Database Queries

Query: “Find all students who have taken all courses offered by the CS department.”

• Let S(x) = “x is a student”
• Let C(y) = “y is a CS course”
• Let T(x, y) = “x has taken y”
• Translation: ∀x (S(x) → ∀y (C(y) → T(x, y)))

Software Specification

Requirement: “Every user should have access to at least one file in every folder.”

• Let U(x) = “x is a user”
• Let Fold(y) = “y is a folder”
• Let File(z) = “z is a file”
• Let In(z, y) = “z is in folder y”
• Let Access(x, z) = “x has access to z”
• Translation: ∀x (U(x) → ∀y (Fold(y) → ∃z (File(z) ∧ In(z, y) ∧ Access(x, z))))

Key Points for Exams

1. Order matters with different quantifiers (∀∃ ≠ ∃∀)
2. Read from left to right, respecting scope
3. To negate: swap each quantifier and negate the predicate
4. Same quantifiers can be reordered (∀∀ or ∃∃)
5. ∀ usually pairs with →, ∃ usually pairs with ∧
6. Draw truth tables for finite domains to verify
7. Practice translating both directions (English ↔ Logic)
8. Watch for hidden quantifiers in English (“all”, “some”, “every”)

Practice Problems

1. What’s the difference between ∀x ∃y (x < y) and ∃y ∀x (x < y)?

2. Negate: ∀x ∃y (x + y > 0)

3. Let domain = {1, 2, 3}. Evaluate: ∃x ∀y (x ≤ y)

4. Translate: “Every even number greater than 2 is the sum of two primes.” (Goldbach’s Conjecture)

5. Translate: “For every student, there is a course they haven’t taken.”

6. True or False: ∀x ∀y P(x, y) ≡ ∀y ∀x P(x, y)?

7. Negate: ∃x ∀y ∃z ((x < y) → (y < z))

8. Express using quantifiers: “There is no largest prime number.”