MATHEMATICS IN THE MODERN WOLRD-ELEMENTARY LOGIC.pptx

ELEM. LOGIC

1. ELEMENTARY LOGIC

2. WHAT IS LOGIC? It allows us to determine the validity of arguments in and out of Mathematics It is a science or discipline that deals with the correct way of reasoning. ARISTOTLE – the father of Logic, the first person who organized the study of logic. Logic comes from the Greek word “logos” which means speech and reasoning.

3. A proposition is a declarative statement that is, by itself, either true or false “but not both” Ex. The sun rise in the east every night. TYPES OF PROPOSITION Simple – means single idea statement. Compound – conveys two or more ideas can be created using logical connectives PROPOSITION CONJUNCTION DISJUNCTION CONDITIONAL BICONDITIONAL NEGATION

4. To every proposition is assigned a truth value . A true proposition has a truth value “true” and a False proposition has a truth value “false”. Sometimes, the symbols T or 1 are used for true propositions and F or 0 are assigned to false propositions. Typically, to denote a proposition, we shall use lower case letters such as p, q, or r. These are called propositional variables or sentential variables. When a sequence of letter and/or logical connectives are given such that when the variables are replaced by specific sentences, a proposition is formed, then we call these sequence of symbols as sentential form of the proposition. PROPOSITION

5. To define a proposition, say p, we usually write: p: <given statement> For instance, p: the earth has two moons. q: seven divides 21. t: Melissa drives a pink D-max. PROPOSITION

6. LOGICAL OPERATORS/CONNECTIVES

7. Consider the following sentences. Determine if the sentences are declarative or not and identify the truth value . ______________ 1. 2 is the only even prime number. ______________ 2. 8 is a multiple of 10. ______________ 3. Mothers are proud of their children. ______________ 4. Are you happy? ______________ 5. x + 3 = 0 ______________ 6. Clean the room before you go. ______________ 7. It will rain today. ______________ 8. Happy birthday! LETS TRY!!! Declarative - True Declarative - False Declarative Not a proposition - Not Declarative – Interrogative Declarative Not a proposition - Not Declarative – Imperative Declarative Not a proposition - Not Declarative – Exclamatory

8. Identify whether the following statements are propositions . If it is a proposition, determine its truth value. p: 3 + 9 = 12 q: 3x – 2 = 13 when x is 5. r: Dogs fly. s: There are 31 days in the month of September. 5. t: is a rational number.   LETS TRY!!! ---- Proposition < True > ---- Proposition < True > ---- Proposition < False > ---- Proposition < False > ---- Proposition < False >

9. Identify whether the following statements are propositions. If it is a proposition, determine its truth value. a: x + 9 = 11 b: y is less than 5 c: Please open the door. d: x(0) = 0 e: c(0) = 1 LETS TRY!!! ---- Not a Proposition ---- Not a Proposition ---- Not a Proposition ---- Proposition < True > ---- Proposition < False > Depends on the value of x

10. TYPES OF PROPOSITION

11. SIMPLE PROPOSITION

12. COMPOUND PROPOSITIONS

13. COMPOUND PROPOSITIONS

14. COMPOUND PROPOSITIONS

15. Charlotte needs to go to work and Paulo is sick. Charlotte needs to go to work but Paulo is sick. Gladys is getting married and Tim is excited. Either the dog is adorable or is playful. The dog is either adorable or is playful. Nikki wants to go to Paris or to London. Nikki wants to go either to Paris or to London.

16. Joan is counting her calories and Jovel doesn’t want to eat dessert. The dog is not adorable and it is playful. The dog is not adorable but it is playful. Charlotte doesn’t needs to go to work or Paulo is not sick. Nikki doesn’t want to go to Paris and she wants to go to London.

17. COMPOUND PROPOSITIONS

18. COMPOUND PROPOSITIONS

19. COMPOUND PROPOSITIONS

20. COMPOUND PROPOSITIONS

21. If it is raining then the sun is not shining. If it is raining then the ground is wet. The ground is wet if and only if it is raining and the sun is not shining.

22. TRUTH VALUE

23. FORMS OF CONDITIONAL STATEMENTS

24. FORMS OF CONDITIONAL STATEMENTS

25. FORMS OF CONDITIONAL STATEMENTS

26. FORMS OF CONDITIONAL STATEMENTS

27. TRANLATE ENGLISH SENTENCE INTO LOGICAL EXPRESSION

28. 1. 10 is not an even integer if and only if 11 is not a prime number. 2. If 11 is prime number then 10 is not an even integer. 3. It is not true that 11 is prime number while 10 is not an even integer. 4. 10 is an even integer if and only if 11 is not a prime. ~p ↔ ~q q → ~p ~q ^ ~p p ↔ ~q

29. (p v q) ^ t p v (q ^ t)

30. TRANLATE ENGLISH SENTENCE INTO LOGICAL EXPRESSION

31. TRANLATE ENGLISH SENTENCE INTO LOGICAL EXPRESSION

32. TRANLATE ENGLISH SENTENCE INTO LOGICAL EXPRESSION