Discrete Mathematics Lecture 3 Applications of Propositional Logic

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Discrete Mathematics Lecture 3: Applications of Propositional Logic and Propositional Equivalences By: Nur Uddin,

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.

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

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

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 2. System specification Example 2: 5

Applications of Propositional Logic 3. Boolean search Google search is the example 6

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

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

Applications of Propositional Logic 5. Logic circuits 8

Propositional Equivalences Definition • A compound proposition that is always true, no matter what

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

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: De Morgan Laws 11

Logical Equivalences Example: 12

Logical Equivalences Example: 12

Logical Equivalences 13

Logical Equivalences 13

Logical Equivalences 14

Logical Equivalences 14

Propositional Satisfiability • A compound proposition is satisfiable if there is an assignment of

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

Application of Propositional Satisfiability 16