Lecture 03 Logic and Propositional Calculus Profs. Koike and Yukita

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Presentation transcript:

Lecture 03 Logic and Propositional Calculus Profs. Koike and Yukita Discrete Systems I Lecture 03 Logic and Propositional Calculus Profs. Koike and Yukita

1. You will be familiar with the following notions. IF p THEN q TRUE, FALSE For all, There exists

2. Propositions – rigorous definition

3. Logical operations and compound propositions

Dangerous zone The English word "or" is commonly used in two distinct ways. Exclusive OR: p or q occurs but not both. Exactly one of the two alternatives occurs. Logical OR: at least one of p or q occurs. Logical operations

Negation Logical operations

Truth tables Logical operations

4. Propositions and truth tables

5. Tautology and Contradictions

6. Logical equivalence

7. Algebra of propositions

Algebra of propositions

Algebra of propositions

Algebra of Propositions De Morgan's laws Algebra of Propositions

8. Conditional statements equivalent Conditional Statements

Conditional Statements Contrapositive Conditional Statements

9. Arguments Arguments

Law of detachment Arguments

A fallacy OK indicates fallacy Arguments

Law of Syllogism Arguments

10. Logical implication All of these are equivalent.

11. Propositional functions Quantifiers

Universal Quantifier Quantifiers

Existential Quantifier Quantifiers

12. Negation of quantified statements negate All math majors are male. It is not the case that all math majors are male. There exists at least one math major who is female(not male). equivalent Negation

De Morgan's laws Negation

Counter example Negation

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9 Negate each of the following statements: All students live in the dormitories. All math majors are males. Some students are 25 years old or older.