Day 05Part 03- Discrete Mathematics for computer science- Nested Quantifiers and its Application

Day 05Part 03- Discrete Mathematics for computer science- Nested Quantifiers and its Application

 Day 05 | Part 03 – Discrete Mathematics for Computer Science

Nested Quantifiers and Its Applications

In Discrete Mathematics, quantifiers are used to express logical statements involving variables. When one quantifier is inside the scope of another quantifier, we call them Nested Quantifiers.

 1. What Are Nested Quantifiers?

A nested quantifier is when a logical expression contains multiple quantifiers (∃, ∀) within each other.

Two types of nested quantifiers:
Same type (∀…∀ or ∃…∃) – Multiple universal or existential quantifiers.
Mixed type (∀…∃ or ∃…∀) – One universal and one existential quantifier.

Examples of Nested Quantifiers:

(i) Universal Universal (∀x ∀y P(x, y))

• Statement: “For all students and for all subjects, every student passes every subject.”
• Mathematical Form: $$\forall x\forall yP(x,y)$$ (Where P(x, y) means “Student x passes subject y”)

(ii) Existential Existential (∃x ∃y P(x, y))

• Statement: “There exists a student and a subject such that the student passes the subject.”
• Mathematical Form: $$\exists x\exists yP(x,y)$$

(iii) Universal Existential (∀x ∃y P(x, y))

• Statement: “For every student, there exists a subject that they pass.”
• Mathematical Form: $$\forall x\exists yP(x,y)$$
• Example in Real Life: Every person has at least one friend.

(iv) Existential Universal (∃x ∀y P(x, y))

• Statement: “There exists a student who passes every subject.”
• Mathematical Form: $$\exists x\forall yP(x,y)$$
• Example in Real Life: “There is a doctor who can treat all patients.”

 2. Applications of Nested Quantifiers

 Computer Science Applications

Database Queries: Nested quantifiers are used in SQL for conditions like “For all customers, there exists an order.”
Algorithm Design: Used in defining properties of sorting, searching, and optimization problems.
Artificial Intelligence: Logical statements in AI systems use quantifiers for reasoning.

 Mathematical Theorems & Proofs

Prime Numbers: ∀n (n is prime → ∀d (if d divides n, then d = 1 or d = n))
Graph Theory: ∃ a Hamiltonian path in a graph G such that ∀ nodes v, v is visited.

 Real-World Applications

Cybersecurity: Every user has at least one authentication method.
Networking: Every device in a network can reach at least one server.

 3. Translating English Sentences to Logical Statements

“Every website has at least one registered user.”

$$\forall w\exists u(R(w,u))$$

“There exists a person who knows everyone in the group.”

$$\exists p\forall g(K(p,g))$$

 Summary

Nested quantifiers help in defining logical relations between multiple variables.
They are essential in databases, algorithms, AI, and mathematical proofs.
Understanding quantifier order is important as it changes the meaning of statements.

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Day 05Part 03- Discrete Mathematics for computer science- Nested Quantifiers and its Application

Nested Quantifiers

Nested Quantifiers

Day 05 – Part 03: Discrete Mathematics for Computer Science

Topic: Nested Quantifiers and Its Application

(For Class / GATE / Computer Science Students)

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1. Quantifiers Recap

In predicate logic (first-order logic), quantifiers are used to express statements over domains.

Two Main Quantifiers:

• Universal Quantifier (∀) – “For all”

  • Example: ∀x P(x) → For all x, P(x) is true.

• Existential Quantifier (∃) – “There exists”

  • Example: ∃x P(x) → There exists an x such that P(x) is true.

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2. What Are Nested Quantifiers?

When quantifiers appear inside one another, we call them nested quantifiers.

General Form:

The order of quantifiers matters!

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3. Examples & Interpretation

➤ Example 1:

Meaning: For every x, there exists a y such that their sum is 0.
True (Let y = -x for any x)

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➤ Example 2:

Meaning: There exists a y such that for all x, x + y = 0.
False (No single y satisfies this for all x)

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Order of Quantifiers – Why It Matters

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4. Nested Quantifiers in Computer Science

Use Cases:

1. Algorithm Correctness

• Example: “For every input, there exists an output such that algorithm terminates.”

  • ∀input ∃output (Correct(input, output))

2. Database Queries (SQL Logic)

• Nested SELECT queries map to nested quantifiers.

3. Formal Verification / Logic Programming

• Proving properties about programs and logical inference (e.g., in Prolog)

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5. Truth Tables & Evaluation

For finite domains, you can evaluate nested quantifiers using truth tables:

Let domain = {1, 2}

Example:

Check:

• x = 1 → ∃y: 1 + y = 3 → y = 2

• x = 2 → ∃y: 2 + y = 3 → y = 1

Statement is true for all x in {1,2}

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6. Practice Questions

1. Determine the truth value:

   • ∀x ∃y (x × y = 1) in integers

2. Translate into English:

   • ∃x ∀y (x ≥ y)

3. Rewrite using logic:

   • “Every person has a friend”

4. Change quantifier order and analyze:

   • ∃x ∀y P(x, y) vs ∀y ∃x P(x, y)

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7. Summary Chart

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Day 05Part 03- Discrete Mathematics for computer science- Nested Quantifiers and its Application

Discrete Mathematics and Its Applications, Eighth Edition