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

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

3. 2. Propositions – rigorous definition

4. 3. Logical operations and compound propositions

5. 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

6. Negation
Logical operations

7. Truth tables
Logical operations

8. 4. Propositions and truth tables

9. 5. Tautology and Contradictions

10. 6. Logical equivalence

11. 7. Algebra of propositions

12. Algebra of propositions

13. Algebra of propositions

14. Algebra of Propositions
De Morgan's laws
Algebra of Propositions

15. 8. Conditional statements
equivalent
Conditional Statements

16. Conditional Statements
Contrapositive
Conditional Statements

17. 9. Arguments
Arguments

18. Law of detachment
Arguments

19. A fallacy OK
indicates fallacy
Arguments

20. Law of Syllogism
Arguments

21. 10. Logical implication
All of these are equivalent.

22. 11. Propositional functions
Quantifiers

23. Universal Quantifier
Quantifiers

24. Existential Quantifier
Quantifiers

25. 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

26. De Morgan's laws
Negation

27. Counter example
Negation

28. Problem 1

29. Problem 2

30. Problem 3

31. Problem 4

32. Problem 5

33. Problem 6

34. Problem 7

35. Problem 8

36. 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.