2024-Handout1 Theory of Numbers Mscied.pptx

Theory of numbers

1. Handout 1: PROPOSITIONAL LOGIC Part 1

2. Definition 2 . Laws of logic are rules of procedures which are used to determine whether a set of statements is true or false. Definition 3 . The truth value of a statement refers to the truthfulness or falsity of the statement. We denote t for the truth value True and f for the truth value False.

3. For definiteness, the assumptions regarding statements are as follows: Assumption 1 : The Law of Excluded Middle For every statement p, either p is true or p is false. Assumption 2: The Law of Contradiction For every statement p, it is not the case that p is both true and false.

4. Definition 4 . Composite statements are statements composed of sub-statements and various connectives. Example 1 . 2 . Refer to Example 1.1. The statement s: 1 is prime and odd is a composite statement. Sub statements: 1. 1 is prime. 2. 1 is odd.

5. Example 1.3: Refer to Example 1.1 and form the following: 1. p ʌq 2. pʌr 3. qʌr

6. Example 1.4. Determine the truth values of the statements in Example 1.3.

7. Example 1.5. Refer to Example 1.1 and form the following: 1. pvq 2. pvr 3. qvr

8. Example 1.5. Give the truth value of each of the disjunctions in Example 1.4.

9. Negation, ~p

10. Example 1.9. Write the truth value of the conditionals in Example 1.8.

11. Example 1.11. Give the truth values of the biconditionals formed in Example 1.10.

12. Seatwork 1

13. DERIVED CONDITIONALS ( Converse, Inverse, Contrapositive) Part 2 Gina A. Malacas

14. Department of Mathematics and Statistics 13 Conditional (Review)

15. Department of Mathematics and Statistics 14 Conditional (Review)

16. Department of Mathematics and Statistics 15 Conditional

17. Department of Mathematics and Statistics 16 Derived Conditional

18. Truth Table for the derived conditionals p q ~ p ~ q Given p →q Converse q →p Inverse ~ p→~q Contrapositive ~ q→~p

19. Example: Given: If 2+5<10 then 2 is an odd number. A) Write the following: 1. Converse 2. Inverse 3. Contrapositive B) Give the truth value of each of the statements in A)