Day 05Part 01-Discrete mathematics-First order Predicate logic or predicate calculus with example
Day 05Part 01-Discrete mathematics-First order Predicate logic or predicate calculus with example
Contents
- 0.1 Discrete Mathematics – First Order Predicate Logic (Predicate Calculus) | Day 05 Part 01
- 0.2 What is Predicate Logic?
- 0.3 Components of Predicate Logic:
- 0.4 Example of Predicate Logic:
- 0.5 Example of Existential Quantifier:
- 0.6 Negation in Predicate Logic:
- 0.7 Applications of Predicate Logic:
- 0.8 Practice Problems:
- 0.9 Day 05Part 01-Discrete mathematics-First order Predicate logic or predicate calculus with example
- 0.10 I. Practice in 1st-order predicate logic – with answers.
- 0.11 Discrete Mathematics Introduction to First-Order Logic
- 1 📘 Day 05 Part 01 – First Order Predicate Logic (FOPL)
- 1.1 🔹 What is Predicate Logic?
- 1.2 ✅ Basic Terms:
- 1.3 🔹 Two Quantifiers in Predicate Logic
- 1.4 🔎 Example 1 – Simple Predicate
- 1.5 🔎 Example 2 – Using ∃ and ∀
- 1.6 🔎 Example 3 – With Two Variables
- 1.7 🔁 Negation of Quantifiers
- 1.8 🧠 Predicate Logic in Real-Life Reasoning
- 1.9 📌 Summary Table
- 1.10 Day 05Part 01-Discrete mathematics-First order Predicate logic or predicate calculus with example
- 1.11 First-Order Logic (First-Order Predicate Calculus)
- 1.12 Discrete Mathematics – Predicates
Discrete Mathematics – First Order Predicate Logic (Predicate Calculus) | Day 05 Part 01
What is Predicate Logic?
Predicate Logic (also known as First-Order Logic or Predicate Calculus) is an extension of propositional logic. It allows expressions with quantifiers and variables, providing more expressive power.
Components of Predicate Logic:
- Predicates: Functions that return True or False based on inputs.
- Example: P(x)P(x): “x is a student.”
- Variables: Represent entities in the domain.
- Example: x,y,zx, y, z
- Quantifiers: Indicate the scope of variables.
- Universal Quantifier ( ∀ ): Means “for all.”
- Example: ∀x P(x) — “For all x, P(x) is true.”
- Existential Quantifier ( ∃ ): Means “there exists.”
- Example: ∃x P(x) — “There exists an x such that P(x) is true.”
- Universal Quantifier ( ∀ ): Means “for all.”
- Logical Connectives:
- ∧ (AND), ∨ (OR), ¬ (NOT), → (IMPLIES), ↔ (IF AND ONLY IF)
Example of Predicate Logic:
- Statement: “All humans are mortal.”
- Predicate Representation:
- Let H(x)H(x): “x is a human.”
- Let M(x)M(x): “x is mortal.”
- Logical Form: ∀x (H(x) → M(x))
- Interpretation: For every x, if x is a human, then x is mortal.
Example of Existential Quantifier:
- Statement: “There exists a student who is intelligent.”
- Predicate Representation:
- S(x)S(x): “x is a student.”
- I(x)I(x): “x is intelligent.”
- Logical Form: ∃x (S(x) ∧ I(x))
- Interpretation: There exists an x such that x is a student and x is intelligent.
Negation in Predicate Logic:
- Negating Universal Quantifier:
- ¬(∀x P(x)) ↔ ∃x ¬P(x)
- Negating Existential Quantifier:
- ¬(∃x P(x)) ↔ ∀x ¬P(x)
Example:
- “Not all birds can fly.”
- B(x)B(x): “x is a bird.”
- F(x)F(x): “x can fly.”
- Logical Form: ¬∀x (B(x) → F(x))
- Equivalent: ∃x (B(x) ∧ ¬F(x)) — “There exists a bird that cannot fly.”
Applications of Predicate Logic:
- Computer Science: Database querying and AI.
- Mathematics: Formal proofs and set theory.
- Linguistics: Analyzing sentence structures.
Practice Problems:
- Express the following statements in predicate logic:
- (a) “Every person loves pizza.”
- (b) “There is a number that is greater than 10.”
- Write the negation of the statement: “All dogs are loyal.”
If you need help solving the problems or understanding the concepts further, feel free to ask!
Day 05Part 01-Discrete mathematics-First order Predicate logic or predicate calculus with example
I. Practice in 1st-order predicate logic – with answers.
Discrete Mathematics Introduction to First-Order Logic
Sure! Let’s explore First Order Predicate Logic (FOPL), also known as Predicate Calculus, with clear explanations and examples — as typically introduced on Day 5 of a Discrete Mathematics course.
📘 Day 05 Part 01 – First Order Predicate Logic (FOPL)
🔹 What is Predicate Logic?
Predicate Logic is an extension of Propositional Logic.
Unlike propositional logic, which deals with simple, atomic statements (e.g., “It is raining”), predicate logic deals with statements that include variables and quantifiers.
✅ Basic Terms:
| Term | Meaning |
|---|---|
| Predicate | A property or relation expressed about a subject (e.g., P(x): “x is tall”) |
| Variable | A symbol like x, y, z representing an element from a domain |
| Quantifier | Symbol indicating “for all” (∀) or “there exists” (∃) |
| Domain | The set of all possible values a variable can take |
🔹 Two Quantifiers in Predicate Logic
-
Universal Quantifier (∀):
Means “for all” or “every”
Symbol:∀x P(x)→ P(x) is true for all values of x -
Existential Quantifier (∃):
Means “there exists”
Symbol:∃x P(x)→ There exists at least one x for which P(x) is true
🔎 Example 1 – Simple Predicate
Let:
-
Domain: All humans
-
Predicate:
M(x): "x is mortal"
Then: -
Statement: “All humans are mortal”
→ Symbolic form:∀x M(x)
🔎 Example 2 – Using ∃ and ∀
Let:
-
Domain: All integers
-
Predicate:
E(x): "x is even"
Then: -
Statement: “There exists an even number”
→ Symbolic form:∃x E(x) -
Statement: “All numbers are even”
→ Symbolic form:∀x E(x)(This is false!)
🔎 Example 3 – With Two Variables
Let:
-
Domain: All people
-
Predicate:
L(x, y): "x loves y"
Then:
-
Statement: “Everyone loves someone”
→∀x ∃y L(x, y) -
Statement: “There is someone whom everyone loves”
→∃y ∀x L(x, y)
🧠 Note: These two statements look similar but are logically very different.
🔁 Negation of Quantifiers
| Statement | Negation |
|---|---|
∀x P(x) |
∃x ¬P(x) |
∃x P(x) |
∀x ¬P(x) |
Example:
-
Original: All students are hardworking →
∀x H(x) -
Negation: Some students are not hardworking →
∃x ¬H(x)
🧠 Predicate Logic in Real-Life Reasoning
-
“If a number is prime, then it is odd”
→ LetP(x): x is prime,O(x): x is odd
→∀x (P(x) → O(x))(⚠️ Note: This is false because 2 is prime and even)
📌 Summary Table
| Symbol | Meaning |
|---|---|
P(x) |
Predicate |
∀ |
Universal Quantifier |
∃ |
Existential Quantifier |
¬ |
NOT |
∧ |
AND |
∨ |
OR |
→ |
IMPLIES |
Would you like:
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📝 Practice problems?
-
🎥 A recommended video lecture in Hindi or English?
-
📄 A PDF summary of this?
Let me know and I can provide it!