Logic-2.pptx (iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii)

very knowledgable in the fields of logic

1. LOGIC

2. Language is .. . Is a systematic way of communicating by the use of sounds of conventional symbols A code human use as a form of expressing themselves and communicating with others A system of words used in particular discipline

3. Mathematics is .. . Is a system of communication about objects like numbers, variables, sets, operations, functions and equation A collection of symbols and their meaning shared by a global community of people who have an interest in the subject A universal language shared by all human being regardless of culture, religion or gender

4. Elements of Mathematical Language has nouns, pronouns, verbs and sentences has own vocabulary, grammar, syntax, synonyms, sentence structure, negations, paragraph structure, conventions and abbreviations specially design so that one can write about numbers, sets, functions, equations, etc. as well as processes undergone by these elements

5. Characteristics of the Language of Mathematics Precise able to make very fine distinctions 2. Concise - able to say thoughts briefly 3. Powerful - able to express complex thoughts with relative ease

6. Logic Statements Every language contains different types of sentences, such as statements, questions and commands. Example “ Is the test today ?” is a question. “ Go get the newspaper ” is a command. “ This is a nice car ” is an opinion. “ Tacloban is the capital of Eastern Visayas ” is a statement of fact.

7. A statement A statement is a declarative sentence that is either true or false. But not both true and false. Example: Eastern Samar State University is located in Borongan .

8. Determine whether each sentence is a statement. Llorente is municipality in Eastern Samar. How are you? 9 9 + 2 is a prime number. x + 1 = 5

9. Simple Statement and Compound Statement A simple statement is a statement that conveys a single idea. A compound statement is a statement that conveys two or more ideas. Connecting simple statements with words and phrases such as and , or , if…then , and if and only if creates a compound statement. Example: “I will attend the meeting or I will go to school.”

10. Determine the simple statements in each compound statement. The principal will attend the class on Tuesday or Wednesday. 5 is an odd number and 6 is an even number. A triangle is an acute triangle if and only if it has three acute angles.

11. George Boole used symbols such as p, q, r, and s to represent simple statements and the symbol Λ , V, ~, →, ↔ to represent connectives. LOGIC CONNECTIVES AND SYMBOLS Statement Connective Symbolic form Type of statement not p not ~ p negation p and q and p Λ q conjunction p or q Or p V q disjunction If p, then q If…then p → q conditional p if and only if q if and only if p ↔ q bi-conditional

12. Unary Operator negation Example: Today is Friday. Negation: Today is not Friday TRUTH TABLE FOR ~ p p ~ p T F F T

13. Write the negation of each statement 1. Ellie Goulding is an opera singer. Ans. Ellie Goulding is not an opera singer. 2. The dog does not need to be feed. Ans. The dog needs to be feed. 3. The Queen Mary 2 is the world’s largest cruise ship. Ans. The Queen Mary 2 is not the world’s largest cruise ship. 4. The fire engine is not red. Ans. The fire engine is red. 5. Bill Gates has a yacht. Ans. Bill Gates does not have a yacht. 6. Avatar was not selected as best picture at the 82nd Academy Awards ceremony. Ans. Avatar was selected as best picture at the 82nd Academy Awards ceremony.

14. Write the negation of each statement. 1. The giants lost the game. Ans. The giants did not lost the game. 2. The game did not go into overtime. Ans. The game went into overtime. 3. The game was not shown on ABC. Ans. The game was shown on ABC.

15. Write compound statement in symbolic form We will find it useful to write compound statements in symbolic form. Consider the following simple statements. p: Today is Friday. q: It is raining. r: I am going to a movie. s: I am not going to the basketball game. Write the following compound statements in symbolic form. Today is Friday and it is raining. It is not raining and I am going to a movie. I am going to the basketball game or I am going to a movie. If it is raining, then I am not going to the basketball game.

16. Write compound statement in symbolic form We will find it useful to write compound statements in symbolic form. Consider the following simple statements. p: Today is Friday. q: It is raining. r: I am going to a movie. s: 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 a basketball game and I am not going to a movie. I am going to a movie if and only if it is raining. If today is Friday, then I am not going to a movie

17. Translate symbolic statements into English sentences Consider the following statements. p: The game will be played in Atlanta. q: The game will be shown on CBS. r: The game will not be shown on ESPN. s: The Mets are favored to win Write each of the following symbolic statements in words. q Λ p The game will be shown on CBS and the game will be played in Atlanta 2. ~ r Λ s The game will be shown on ESPN and the Mets are favored to win. 3. s ↔ ~ p The Mets are favored to win if and only if the game will not be played in Atlanta.

18. Translate symbolic statements into English sentences Consider the following statements. e: All men are created equal. t: I am trading places. a: I get Abe’s place. g: I get George’s place. Use the above information to translate the dialogue in the speech bubbles below.

19. Quiz no. 3 Determine whether each sentence is a statement. Star Wars: The Force Awakens is the greatest movie of all time. The area code for Storm Lake, Iowa, is 512. Have a fun trip. Mickey Mouse was the first animated character to receive a star on the Hollywood walk of fame.

20. Quiz no. 3 Write each symbolic statement in words. Use p, q, r, s, t, and u as defined below. p: The tour goes to Italy. q: The tour goes to Spain. r: We go to Venice. s: We go to Florence. t: The hotel fees are included. u: The meals are not included. 5. p Λ ~q 8. p → r 6. r V s 9. s ↔ ~ r 7. r → ~s 10. ~t Λ u

21. Compound statements and Grouping Symbols If compound statement is written in symbolic form, then parentheses are used to indicate which simple statements are grouped together. The table below illustrates the use of parentheses to indicate groupings for some statements in symbolic form. Symbolic Form The parentheses indicate that: p Λ (q V ~ r) q and ~ r are grouped together. (p Λ q) V r p and q are grouped together. (p Λ ~ q) → (r V s) p and ~ q are grouped together. r and s are grouped together.

22. Compound statements and Grouping Symbols If compound statement is written as English sentence, then a comma is used to indicate which simple statements are grouped together. Statements on the same side of a comma are grouped together. English sentence The comma indicates that: p, and q or not r. q and ~ r are grouped together because they are both on the same side of he comma. p and q, or r. p and q are grouped together because they are both on the same side of he comma. If p and not q, then r or s. p and ~ q are grouped together because they are both to the left of he comma. r and s are grouped together because they are both to the right of the comma.

23. Example Translate compound statements Let p, q, and r represent the following. p: You get a promotion. q: You complete the training. r: You will receive a bonus. 1. Write (p Λ q) → r as an English sentence Ans. If you get a promotion and you complete the training, then you will receive a bonus. 2. If you do not complete the training, then you will not get the promotion and you will not receive a bonus. Ans. ~q → (~p Λ ~ r)

24. Example Translate compound statements Let p, q, and r represent the following. p: Thelma’s singing style is similar to Lina. q: Thelma has messy hair. r: Thelma is a rapper. Write (p Λ q) → r as an English sentence Ans. If Thelma’s singing style is similar to Lina and Thelma has messy hair, then Thelma is a rapper. 2. Write “ If Thelma is not a rapper, then Thelma does not have messy hair and Thelma’s singing style is not similar to Lina .” in symbolic form Ans. ~r → (~q Λ ~ p)

25. Truth Value of Conjunction The conjunction p Λ q is true if and only if both p and q are true. False if otherwise. Symbol: Λ Meaning: “and” Example: p Λ q Meaning: “p and q” Example: Today is Friday and a Holiday. p: Today is Friday. q: Today is holiday. Conjunction: p Λ q

26. Truth Value of Conjunction Truth table for p Λ q p q p Λ q T T T F F T F F T F F F

27. Truth Value of Disjunction The disjunction p V q is true if and only if p is true, q is true or both p and q are true. Symbol: V Meaning: “or” Example: p V q Meaning: “p or q” Example: Today is Friday, or I am staying at home.. p: Today is Friday. q: I am staying at home. Conjunction: p Λ q

28. Truth Value of Disjunction Truth table for p V q p q p V q T T T F F T F F T T T F

29. Determine the truth value of the statement Determine whether the statement is true or false. 1) 7 ≥ 5 Ans. true 2) 5 is a whole number and 5 is an even number. Ans. false 3) 2 is a prime number and 2 is an even number. Ans. true 4) 21 is a rational number and 21 is a natural number. Ans. true 5) 4 ≤ 9. Ans. true 6) -7 ≥ -3. Ans. false

30. QUANTIFIERS AND NEGATION EXISTENTIAL QUANTIFIERS – are used as prefixes to assert the existence of something. (Examples are: some , there exist , at least one) UNIVERSAL QUANTIFIERS – none and no deny the existence of something , whereas the universal quantifiers all and every are used to assert that every element of a given set satisfies some condition.

31. Example What is the negation of the following statements? 1.) “no doctors write in a legible manner.” 2.) “all doctors write in a legible manner.” ANSWER: 1) “some doctors write in a legible manner.”

32. Sometimes, the negation of a quantified statement must be considered The following illustrates how to write the negation of some quantified statements QUANTIFIED NEGATION STATEMENT “Some…are” “no…are” “All…are” “Some...are not” “No…are” “Some…are” “Some…are not” “all…are”

33. Some roses are red. Ans. No roses are red.

34. Sometimes, the negation of a quantified statement must be considered The following illustrates how to write the negation of some quantified statements QUANTIFIED NEGATION STATEMENT “Some…are” “no…are” “All…are” “Some...are not” “No…are” “Some…are” “Some…are not” “all…are”

35. All violets are blue . Answer: Some violets are not blue.

36. Sometimes, the negation of a quantified statement must be considered The following illustrates how to write the negation of some quantified statements QUANTIFIED NEGATION STATEMENT “Some…are” “no…are” “All…are” “Some...are not” “No…are” “Some…are” “Some…are not” “all…are”

37. No violets are blue Answer: Some violets are blue. (there exists a violet that is blue)

38. Sometimes, the negation of a quantified statement must be considered The following illustrates how to write the negation of some quantified statements QUANTIFIED NEGATION STATEMENT “Some…are” “no…are” “All…are” “Some...are not” “No…are” “Some…are” “Some…are not” “all…are”

39. Some Roses are not red. Answer: All roses are red.

40. Example of quantified statement and their negations NEGATION All X are Y. Some X are not Y. No X are Y. Some X are Y.

41. Examples Some lions are playful. Negation: No lions are playful . Some dogs are not friendly. Negation: All dogs are friendly . 3. All bears are brown. Negation: Some bears are not brown . 4. No smart phones are expensive. Negation: Some smart phones are expensive .

42. More examples Some airports are open. Negation: No airports are open. All movies are worth the price of admission. Negation: Some movies are not worth the price of admission. 3. Some doctors are not rich. Negation: All doctors are rich . 4. All cars run on gasoline Negation: Some cars do not run on gasoline.

43. Quiz Determine whether each statement is true or false. 1) 7 < 5 or 3 > 1. 2) 3 ≤ 9. 3) (-1) 50 = 1 and (-1) 99 = -1. 4) 7 ≠ 3 or 9 is a prime number. 5) -5 ≥ -11. 6) 4 .5 ≤ 5.4. 7) 2 is an odd number or 2 is an even number.

44. Write the negation of each quantified statement. Start each negation with “Some,” “No,” or “All.” 8) Some lions are playful. 9) Some dogs are not friendly. 10) All classic movies were first produced in black and white. 11) No even numbers are odd numbers. 12) Some actors are not rich. 13) All cars run on gasoline. 14) None of the students took my advice.

45. Truth Tables, Equivalent Statements, and Tautologies p ~p T F F T Negation p q p Λ q T T T F F T F F T F F F Conjunction Disjunction p q p V q T T T F F T F F T T T F

46. Truth Tables, Equivalent Statements, and Tautologies Standard truth table form for a given statement that involves only the two simple statement p and q. p q Given statement T T T F F T F F

47. Example: 1. Construct a truth table for ~(~p V q) V q 2. Use the truth table above to determine the truth value of ~(~p V q) V q, given that p is true and q is false. Ans. True (T) p T T F F q T F T F ~ p F F T T ~p V q T F T T ~(~p V q) F T F F ~(~p V q) V q T T T F

48. Example: 2. Construct a truth table for (p Λ ~ q) V (~p V q) p T T F F q T F T F ~ p F F T T ~q F T F T (p Λ ~q) F T F F (~p V q) T F T T (p Λ ~q) V (~p V q) T T T T

49. Example: 2. Use the truth table that you constructed in no. 2 to determine the truth value of (p Λ ~q) Λ (~p V q), given that p is true and q is false. p T T F F q T F T F ~ p F F T T ~q F T F T (p Λ ~q) F T F F (~p V q) T F T T (p Λ ~q) V (~p V q) T T T T

50. Truth Tables, Equivalent Statements, and Tautologies Compound statement that involves exactly three simple statements require a standard truth table form with 2 3 = 8 rows, as shown below p q r Given statement T T T T T F T F F T F T F F F F