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AI & Data Mining/Week 21/Week 21 - Transformational Proofs in Propositional Logic.md

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+**Notes on Slides and Exercises**
+
+**Slide 0: Learning Objectives**
+- Prove equivalence of formulae using truth tables.
+- Remember and use laws of equivalence.
+- Carry out a transformational proof.
+
+**Slide 1: Contents**
+- Why is propositional logic called Boolean logic/algebra?
+- Using truth tables to prove two formulae are identical in meaning.
+- A problem with truth tables.
+- Laws of logic.
+- Famous applications.
+- Summary, reading and references.
+
+**Slide 2: George Boole (1815-1864)**
+- Son of a shoemaker, self-taught mathematician.
+- Professed mathematics at Queens College, Cork, Ireland.
+- Seminal work: attempted to apply algebraic and arithmetic principles to logic.
+
+**Slide 3: Logical Equivalence (≡)**
+- Two formulae are equivalent if they have identical truth values under all possible assignments.
+- Example: p ≡ p ∨ p (Idempotence)
+
+**Slide 4: Two Approaches to Logical Equivalence**
+1. Truth tables
+   - Example: p ≡ ¬p (Negation)
+2. Transformational proofs
+   - Uses laws of logic to prove equivalence.
+
+**Slide 5: Logical Equivalence with One Variable**
+- Truth table for p ≡ p ∨ p (Idempotence)
+  - T | T | T
+    F | F | F
+
+**Slide 6: Logical Equivalence with Two Variables**
+- Truth table for p ∨ q ≡ q ∨ p (Commutativity)
+  - T | T | T | T
+    F | T | T | T
+    T | F | T | F
+    F | F | F | F
+
+**Slide 7: Exercise - De Morgan's First Law**
+- Prove ¬(p ∧ q) ≡ ¬p ∨ ¬q using a truth table.
+
+**Slide 8: Solution to Exercise**
+- Truth table for ¬(p ∧ q) ≡ ¬p ∨ ¬q
+  - T | F | F | T | F | F | T | F | F | F
+    F | F | F | F | F | F | F | F | F | T
+
+**Slide 9: Logical Equivalence with Three Variables**
+- Truth table for (p ∨ q) ∨ r ≡ p ∨ (q ∨ r) (Associativity)
+- Example: (2 × 3) × 4 = 2 × (3 × 4)
+
+**Slide 10: A Problem with Truth Tables**
+- Truth tables become increasingly tedious as the number of variables increases.
+  - n | Number of rows in truth table (2^n)
+    1 | 2
+    2 | 4
+    3 | 8
+
+**Slide 11: Laws of Logic**
+- Idempotence: p ≡ p ∨ p, p ≡ p ∧ p
+- Commutativity: p ∨ q ≡ q ∨ p, p ∧ q ≡ q ∧ p
+- Associativity: (p ∨ q) ∨ r ≡ p ∨ (q ∨ r), etc.
+- Negation law: ¬¬p ≡ p
+- Law of equivalence: p ⇔ q ≡ (p ⇒ q) ∧ (q ⇒ p)
+- Law of implication: p ⇒ q ≡ ¬p ∨ q
+- Contraposition Law: p ⇒ q ≡ ¬q ⇒ ¬p
+- De Morgan's first law: ¬(p ∧ q) ≡ ¬p ∨ ¬q
+- De Morgan's second law: ¬(p ∨ q) ≡ ¬p ∧ ¬q
+
+**Slide 12: Proof of De Morgan's Second Law**
+- Using laws of logic to prove ¬(p ∨ q) ≡ ¬p ∧ ¬q.
+
+**Slide 13: Idempotence**
+- Proof of p ∧ p ≡ p using laws of logic.
+  - p ∧ p ≡ ¬¬p ∧ ¬¬p (Negation law twice)
+  - ≡ ¬(¬p ∨ ¬p) (De Morgan's second law)
+  - ≡ ¬false (Law of contradiction)
+  - ≡ p (Simplification)
+
+**Slide 14: Commutative Laws of Logic**
+- Proof of p ∧ q ≡ q ∧ p using laws of logic.
+  - p ∧ q ≡ ¬¬p ∧ ¬¬q (Negation law twice)
+  - ≡ ¬(¬p ∨ ¬q) (De Morgan's second law)
+  - ≡ ¬(¬q ∨ ¬p) (Commutativity of ∨)
+  - ≡ q ∧ p (Negation law twice)
+
+**Slide 15: Associative Laws of Logic**
+- Example: (a × b) × c ≡ a × (b × c)
+
+**Slide 16: Distributive Laws of Logic**
+- Example: a × (b + c) ≡ (a × b) + (a × c)
+
+**Slide 17: T versus true and F versus false**
+- T represents a formula is true, F represents it is false.
+- Example: p ∧ true ≡ p (Law of simplification)
+
+**Slide 18: Laws involving true and false**
+- Law of excluded middle: p ∨ ¬p ≡ true
+- Law of contradiction: p ∧ ¬p ≡ false
+- Laws of simplification: p ∧ true ≡ p, p ∨ true ≡ true, p ∧ false ≡ false, p ∨ false ≡ p
+
+**Slide 19: Two More Laws of Simplification**
+- Proofs for:
+  - p ∨ (p ∧ q) ≡ p
+  - p ∧ (p ∨ q) ≡ p
+
+**Slide 20: Transformational Proofs**
+- Two implicit rules:
+  - Substitution Rule: Replace a sub-formula with an equivalent one without changing the meaning.
+  - Transitivity Rule: If p ≡ q and q ≡ r, then p ≡ r.
+- Example: Modus Tollens (¬q ∧ (p ⇒ q) ⇒ ¬p)
+  - Proof using laws of logic.
+
+**Slide 21: Exercise**
+- Fill in the missing steps to prove (p ∨ q) ∧ (¬p ∨ q) ≡ q using transformational proofs.
+- Solution:
+  - (p ∨ q) ∧ (¬p ∨ q)
+  - ≡ (q ∨ p) ∧ (¬p ∨ q) (Commutativity)
+  - ≡ q ∨ (p ∧ ¬p) (Distributive law)
+  - ≡ q ∨ false (Law of contradiction)
+  - ≡ q (Simplification)
+
+**Slide 22: Example - Accommodation**
+- Prove ¬p ⇒ (q ∨ r) ≡ (¬p ∧ ¬q) ⇒ r using transformational proofs.
+- Solution:
+  - Proof using laws of logic.
+
+**Slide 23: Famous Applications**
+- Analysis of complex conditional commands in programming.
+- Design of digital circuits.
+- Algebraic approach to formal specifications in software engineering.
+
+**Slide 24: Summary**
+- Boole's system was the foundation for propositional logic.
+- Two approaches to establishing equivalence:
+  - Truth tables (tedious with many variables)
+  - Transformational proofs (uses laws of logic)
+- Summary slides on Boole's system, truth tables, and transformational proofs.
+
+**Slide 25: Reading and References**
+- Recommended textbook: Russell, S., & Norvig, P. (2022). Artificial Intelligence: A Modern Approach (4th ed.). Pearson.
+- Additional references:
+  - Nissanke, M. (1999). Introductory Logic and Sets for Computer Scientists. Addison-Wesley.
+  - Gray, P. (1984). Logic, Algebra and Databases. John Wiley & Sons.
+
+**Exercises:**
+1. Prove the laws of idempotence, commutative, associative, and distributive using truth tables.
+2. Prove De Morgan's laws using truth tables or transformational proofs.
+3. Prove the laws involving true and false using truth tables or transformational proofs.
+4. Prove the laws of simplification using truth tables or transformational proofs.
+5. Prove the equivalence of two given formulae using transformational proofs (as demonstrated in slides 21 and 22).