Discrete Mathematics Lecture 3 Applications of Propositional Logic
- Slides: 16
Discrete Mathematics Lecture 3: Applications of Propositional Logic and Propositional Equivalences By: Nur Uddin, Ph. D 1
Applications of Propositional Logic 1. Translating sentences 2. System specification 3. Boolean search 4. Logic puzzles 5. Logic circuits 2
Applications of Propositional Logic 1. Translating sentences Example: You can access the Internet from campus only if you are a computer science major or you are not a freshman. 3
Applications of Propositional Logic 2. System specification Example 1: The automated reply cannot be sent when the file system is full Example 2: 4
Applications of Propositional Logic 2. System specification Example 2: 5
Applications of Propositional Logic 3. Boolean search Google search is the example 6
Applications of Propositional Logic 4. Logic Puzzles Example: In [Sm 78] Smullyan posed many puzzles about an island that has two kinds of inhabitants, knights, who always tell the truth, and their opposites, knaves, who always lie. You encounter two people A and B. What are A and B if A says “B is a knight” and B says “The two of us are opposite types? ” 7
Applications of Propositional Logic 5. Logic circuits 8
Propositional Equivalences Definition • A compound proposition that is always true, no matter what the truth values of the propositional variables that occur in it, is called a tautology. • A compound proposition that is always false is called a contradiction. • A compound proposition that is neither a tautology nor a contradiction is called a contingency. 9
Logical Equivalences • Compound propositions that have the same truth values in all possible cases are called logically equivalent. • The compound propositions p and q are called logically equivalent if p ↔ q is a tautology. • The notation p ≡ q denotes that p and q are logically equivalent. 10
Logical Equivalences: De Morgan Laws 11
Logical Equivalences Example: 12
Logical Equivalences 13
Logical Equivalences 14
Propositional Satisfiability • A compound proposition is satisfiable if there is an assignment of truth values to its variables that makes it true. • Example: Check satisfiability of the following compound propositions a. (p ∨¬q) ∧ (q ∨¬r) ∧ (r ∨¬p) b. (p ∨ q ∨ r) ∧ (¬p ∨¬q ∨¬r) c. (p ∨ q ∨ r) ∧ (¬p ∨¬q ∨¬r) • When we find a particular assignment of truth values that makes a compound proposition true, we have shown that it is satisfiable; such an assignment is called a solution of this particular satisfiability problem. 15
Application of Propositional Satisfiability 16
- Logical equivalence table
- Discrete math propositional logic
- First order logic vs propositional logic
- First order logic vs propositional logic
- First order logic vs propositional logic
- A computer programming team has 13 members
- What is discrete math
- Negation math
- Propositional logic symbols
- Valid
- Proposition examples sentences
- The proposition ~p ν (p ν q) is a
- Implies in propositional logic
- Simple proposition
- Compound statement symbols
- Pros and cons of propositional logic
- Uniqueness quantifier equivalence