Elementary Mathematical Logic .pptx

Logic

1. “ If it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t . That’s logic.” Elementary Mathematical Logic

2. Table of contents Statement 01 Negation 02 Conjunction 03 Disjunction 04

3. Table of contents Conditional 05 Biconditional 06 Logical Equivalence 07

4. Directions: Identify if the following sentences are STATEMENT or NOT STATEMENT and determine its truth value (that is you should be able to decide whether the statement is TRUE or FALSE). In your activity notebook. (3 minutes) Am I Statement or Not?

5. 1. Happy Birthday! 2. Some books have hard covers. 3. No Philippine presidents were naturally born in Korea. 4. My dog loves me. 5. Some cats aren't mammals. 6. Juan is intelligent. 7. Go get the milk. 8. All dogs are Labrador Retrievers. 9. Can we be friends? 10. Today is Tuesday. 5

6. 1. Happy Birthday! – not statement 2. Some books have hard covers. - statement 3. No Philippine presidents were residents of Korea. - statement 4. My dog loves me. - not statement 5. Some cats aren't mammals- statement 6. Juan is intelligent- not statement 7. Go get the milk.- not statement 8. All dogs are Labrador Retrievers.- statement 9. Can we be friends?- not statement 10. Today is Tuesday.- statement 6

7. 2. Some books have hard covers. – statement - TRUE 3. No Philippine presidents were naturally born in Korea. – statement- TRUE 5. Some cats aren't mammals- statement - FALSE 8. All dogs are Labrador Retrievers.- statement- FALSE 10. Today is Tuesday.- statement-FALSE 7

8. What can you say about the sentences? How did you know that a sentence is a statement or not? How would you define a statement?

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10. I. What is Logic? -is the science of correct reasoning.

11. Statement 01 is any declarative sentence that is either true (T) or false (F), but not both

12. We refer to T or F as the truth value of the statement. Statements are usually denoted by lower case letters (for example: p, q, r,s …)

13. The sentence “Stand up” or “Follow the leader” cannot be used as statements. These are commands instead of a declarative sentence and therefore cannot be classified as true or false

14. Examples of statements: (TRUE) (FALSE) Today is Thursday. s: January has 31 days.   Examples of NOT statements: 𝑥+𝑦 Happy Birthday! Tweet me. (command) Can we be friends? (question) Juan is intelligent. (opinion)

15. Let p be the sentence “Manila is the capital in the Philippines.” Let q be the sentence “The world is flat.” These are both statements because we can assign to them a truth value. In this case, p is true and q is false. Example 1

16. Simple Statement ? Compound Statement ?

17. A simple statement is a statement that conveys a single idea. Examples: Today is March 16, 2020. (Simple statement) Metro Manila is under community quarantine. (Simple statement)

18. A compound statement is a statement that conveys two or more ideas. Examples: Today is March 16, 2020 and Metro Manila is under community quarantine. (Compound statement) Today is not March 16, 2020 or yesterday was a Sunday. ( Compound statement)

19. Statement Connective Symbolic form Type of statement not not negation and and conjunction or or disjunction If , then If … then conditional if and only if if and only if Biconditional Statement Connective Symbolic form Type of statement not negation and conjunction or disjunction If … then conditional if and only if Biconditional Logic Connectives and Symbols M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

20. Negation 02 The negation of a statement p, denoted by ~p (read as “not p”) is the statement whose truth value is the opposite of the truth value of p.

21. Example p= Manila is not the capital city of the Philippines p= Manila is the capital city of the Philippines Negation

22. Activity : Negate me! Write the negation of each statement: 1. I want a car and a motorcycle. 2. My cat stays outside or it makes a mess. 3. I've fallen and I can't get up. 4. You study or you don't get a good grade. 5. It is raining and it isn't snowing.

23. Activity 1: Negate me! ( Answer) Write the negation of each statement: 1. I don’t want a car and a motorcycle. 2. My cat won’t stay outside or it won’t make a mess. 3. I haven’ t fallen and I can get up. 4. You don’t study or you do get a good grade. 5. It is not raining and it is snowing. M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

24. Example: Consider the following simple statements. Today is Friday. It is raining. I am going to a movie. I am not going to the basketball game. Write the following compound statements in symbolic form Today is not Friday and I am going to a movie. I am going to the basketball game and I am not going to a movie. I am going to the movie if and only if it is raining. If today is Friday, then I am not going to a movie.  

25. Consider the following simple statements. Today is Friday. It is raining. I am going to a movie. I am not going to the basketball game. Write the following compound statements in symbolic form. SOLUTION: Today is not Friday and I am going to a movie. Answer: I am going to the basketball game and I am not going to a movie. Answer: I am going to the movie if and only if it is raining. Answer: If today is Friday , then I am not going to a movie. Answer:  

26. Activity 9: Translate me! Direction: Consider the following simple statements. Translate the following symbolic form in compound statements. Today is Friday. It is raining. I am going to a movie. I am not going to love my ex anymore.   M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

27. Today is Friday . It is raining . I am going to a movie. I am not going to love my ex .   1. (~ p q ) 2. (s ∧ ~q) 3. (r ~p) 4. ( s ~q ) 5. (~q ∧ p ) 6. (~p ∨ q) 7. (p ~p) 8. (s ~r) 9. (~r ∨ s ) 10. (~q ∨ s )  

28. Application: Construct me! Directions: Construct your own 4 simple or compound statements labeling it with p, q, r and s respectively. Translate the symbolic form given below using the statements you constructed. 1. (s ∧ ~q) 2. ~(~q ∧ r) 3. ~(p ∨ q) 4. (p ∧ ~q) 5. (~s ∧ q)

29. The truth value of a simple statement is either true (T) or false (F). The truth value of a compound statement depends on the truth values of its simple statements and its connectives.

30. Conjunction , is true if BOTH and are true. Otherwise, it is false.   M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

31. Disjunction , is true if at least one statements ( or both) is true. It is false only if both statements are false.   M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

32. Examples: Negating simple statements. , June 26, 2020 is a Friday. June 26, 2020 is not a Friday. The sun is bigger than the moon. The sun is not bigger than the moon. I am not a Libra. I am a Libra.   Negation, If is true, is false If is false, is true   M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic