LOGIC, REASONING, PROPOSITIONS, Copy.pptx

Logic, arguments, reasoning, and compound propositions are important topics in General Mathematics that deal with correct thinking and decision-making. Logic helps us know if statements are true or false. Arguments use reasons or premises to support a conclusion. Reasoning is the process of thinking clearly to reach a valid conclusion, using either deductive or inductive thinking. Compound propositions combine two or more simple statements using connectives like and, or, not, and if–then. These concepts help us analyze ideas and make sound conclusions.

1. LOGIC General Mathematics

2. TOPICS Propositions Truth Value and Logical Equivalence Categorical Propositions and Validity Deductive Arguments Determining Validity of Arguments using Truth Table Direct Proof and Indirect Proof 2/2/20XX PRESENTATION TITLE 2

3. Introduction In our daily lives, you are faced with numerous issues you need to resolve and decide on, from simple ones such as what clothes to wear or how to update your social network status to more sensitive matters such as what to study in college or which presidential candidate to vote. 2/2/20XX PRESENTATION TITLE 3

4. What is Logic? Logic is traditionally defined as the study of the laws of thought or correct reasoning, and is usually understood in terms of inferences or arguments. Reasoning is the activity of drawing inferences. Arguments are the outward expression of inferences. An argument is a set of premises together with a conclusion. 2/2/20XX PRESENTATION TITLE 4

5. Topic One PROPOSITIONS

6. What is “proposition”? Proposition is a declarative sentence or statement that is either true or false but not both. If the proposition is true, then its truth value is true and if it is false its truth value is false. 2/2/20XX PRESENTATION TITLE 6

7. Then what if the declarative statement is neither be true, nor false or is both true or false? Then that statement is called a paradox . 2/2/20XX PRESENTATION TITLE 7

8. Try it! Proposition or Paradox? This statement is false. All birds can fly. Quezon City is our nation’s capital. Everything I say is a lie. Doctor have bad handwritings. 2/2/20XX PRESENTATION TITLE 8

9. Answer Proposition All birds can fly. Quezon City is our nation’s capital. Paradox This statement is false. Everything I say is a lie. Doctor have bad handwritings. 2/2/20XX PRESENTATION TITLE 9

10. Types of Proposition Simple Proposition A proposition is simple if it is contains only one idea. Compound Proposition A proposition is compound if it is composed of at least two simple propositions joined together by logical connectives. - Negation - Conjunction - Disjunction - Conditional - Biconditional 2/2/20XX PRESENTATION TITLE 10

11. Type of Compound Preposition Logical Operator (Connective) Symbolic Form Read as Negation Not ˜ P Not P Conjunction And (but, also and moreover) P ^ Q P and Q Disjunction Or (unless) P v Q P or Q Conditional If … then P -> Q If P, then Q Biconditional If and only if P <-> Q P if and only if Q 2/2/20XX PRESENTATION TITLE 11

12. Negation Just like what you might expect from a negation, this logical operator states the exact opposite of a given statement. Consider the simple proposition E: e represents an irrational number. The following statements of the different ways ˜E maybe expressed. e does not represent an irrational number. It is not the case that e represents an irrational number. It is false that e represents an irrational number. 2/2/20XX PRESENTATION TITLE 12

13. Conjunction Other words that can be used in conjunction are but, also and moreover. Example: Symbolic Form: D ^ R where D: the domain of f(x) = log x is zero R: the range of f(x) = log x is real number - The domain of f(x) = log x is zero and its range is real number. 2/2/20XX PRESENTATION TITLE 13

14. Example of Conjunction Let p represent the proposition “He has a green thumb” and q represent the proposition “ He is a senior citizen” Answer: He has a green thumb and he is a senior citizen. In symbol: p^q 2/2/20XX PRESENTATION TITLE 14

15. Disjunction Unless is another word used for the disjunction of propositions. The following statements illustrate the different ways a disjunction can be expressed: a=0 or b=0 a=0 unless b=0 Either a or b is equal to zero. 2/2/20XX PRESENTATION TITLE 15

16. Example of Disjunction Let p represent the proposition “He has a green thumb” and q represent the proposition “ He is a senior citizen” Answer: He has a green thumb or he is a senior citizen. In symbol: p v q 2/2/20XX PRESENTATION TITLE 16

17. Conditional Other words that can be used in place of the connective “if…then” are “only if” and “implies”. In P -> Q, you say that P is a sufficient condition for Q while Q is necessary condition for P. Also, you refer to P and Q as the antecedent and the consequent of the implication, respectively. The following statements illustrate the different ways an implication can be expressed in sentence form: If I saved some money, then I will buy some of the things I need. I will buy some of the things only if I will saved some of my money. 2/2/20XX PRESENTATION TITLE 17

18. Conditional From conditional proposition, you can derive its converse, inverse and contrapositive. If P -> Q Then, Converse: Q -> P Inverse: ˜P -> ˜Q Contrapositive: ˜Q -> ˜P 2/2/20XX PRESENTATION TITLE 18

19. Conditional Example: “If you are honest, then you deserve a thumbs-up.” P = You are honest Q = You deserve a thumbs-up Converse: If you deserve a thumbs-up, then you are honest. Inverse: If you are not honest, then you do not deserve a thumbs-up Contrapositive: If you don’t deserve a thumbs-up, then you are not honest. 2/2/20XX PRESENTATION TITLE 19

20. Example of Conditional Let p represent the proposition “He has a green thumb” and q represent the proposition “ He is a senior citizen” Answer: If he has a green thumb, then he is a senior citizen. In symbol: p  q 2/2/20XX PRESENTATION TITLE 20

21. Biconditional If P <-> Q is true, you say P and Q are logically equivalent. That is, they will be true under exactly same circumstances. Example: Symbolic Form: E <-> N where E: the function f is even N: f(x) = f(-x) - The function f is even if and only if f(x) = f(-x).

22. Example of Biconditional Let p represent the proposition “He has a green thumb” and q represent the proposition “ He is a senior citizen” Answer: He has a green thumb if and only if he is a senior citizen. In symbol: p  q 2/2/20XX PRESENTATION TITLE 22

23. Take note The use of commas indicates which simple statements are grouped together. 2/2/20XX PRESENTATION TITLE 23

24. TRANSLATE THE STATEMENT INTO SYMBOLIC FORM. He has a green thumb or he is a senior citizen, and lives in Manila. It is not the case that mother and son love each other. He does have a green thumb or he is not a senior citizen. If a mother loves her son and son loves his mother, then father loves his son. 2/2/20XX PRESENTATION TITLE 24

25. ANSWER KEY (p v q) ^ r ~p p v ~q (p ^ q)  r 2/2/20XX PRESENTATION TITLE 25

26. Take note: In identifying the type of symbolic statements, we follow the dominance of connectiveness. The list below gives the connectives in their dominant order (from the strongest to the weakest) Biconditional Conditional Disjunction/Conjunction Negation 2/2/20XX PRESENTATION TITLE 26

27. Let’s Try! If you are interested in becoming a volunteer, you should fill-up the application form and submit it to our headquarters or at any of our local offices. Let a: You are interested in becoming a volunteer b: You should fill-up the application form c: You should submit it to our headquarters d: You should submit it to any of our local offices Symbolic Form: a -> (b ^ (c v d)) 2/2/20XX PRESENTATION TITLE 27

28. Topic Two TRUTH VALUE AND LOGICAL EQUIVALENCE

29. TRUTH TABLE -The truth (table)(value) of a statement is the classification as true or false which denoted by T or F. -is listing of all possible combinations of the individual statement as true or false, along with the resulting truth value of the compound statements. 2/2/20XX PRESENTATION TITLE 29

30. TRUTH TABLE -Truth tables aide in distinguishing valid and invalid arguments. NUMBER OF ROWS -If a compound statement consists of n individual statements, each represented by a different letter, the number of rows required in the truth table is 2^n 2/2/20XX PRESENTATION TITLE 30

31. Truth Table Negation Conjunction Disjunction 2/2/20XX PRESENTATION TITLE 31 P ˜P T F F T P Q P^Q T T T T F F F T F F F F P Q P v Q T T T T F T F T T F F F

32. Truth Table Conditional Biconditional 2/2/20XX PRESENTATION TITLE 32 P Q P -> Q T T T T F F F T T F F T P Q P <-> Q T T T T F F F T F F F T

33. Let’s Try! Construct the truth table of P ^ ˜P P ˜ P P ^ ˜P T F F F T F 2/2/20XX PRESENTATION TITLE 33 P ^ ˜ P is an example of a contradiction. Contradictions are important because they provide you an alternative approach in establishing the validity or invalidity of a logical argument which we shall discuss in lesson 9. A contradiction is a compound proposition that is false for all possible truth values of its component propositions.

34. Let’s Try! Construct the truth table of (˜P v Q) <-> (P -> Q) 2/2/20XX PRESENTATION TITLE 34 This is an example of a tautology. A tautology proposition that is true for all possible truth values of its component propositions. P Q ˜P ˜P v Q P -> Q (˜P v Q) <-> (P -> Q) T T F T T T T F F F F T F T T T T T F F T T T T

35. Let’s Try! Construct the truth table of P ^ ˜Q -> R 2/2/20XX PRESENTATION TITLE 35 P ^ ˜Q -> R is an example of a contingency. A contingency proposition is a compound proposition that is neither a tautology nor a contradiction. P Q R ˜Q P ^ ˜Q P ^ ˜Q -> R T T T F F T T T F F F T T F T T T T T F F T T F F T T F F T F T F F F T F F T T F T F F F T F T

36. Logical Equivalence Two propositions are logically equivalent if they have same truth values. 2/2/20XX PRESENTATION TITLE 36

37. Let’s Try! Construct the truth table of ˜Q -> ˜ P 2/2/20XX PRESENTATION TITLE 37 Compare the truth value of ˜Q -> ˜ P to P -> Q they have the same result. This equivalence is known as contraposition. P Q ˜Q ˜ P ˜Q -> ˜ P T T F F T T F T F F F T F T T F F T T T

38. Let’s Try! Construct the truth table of (P -> Q) ^ (Q -> P) 2/2/20XX PRESENTATION TITLE 38 Compare it to P <-> Q they have same truth values. P Q (P -> Q) (Q -> P) (P -> Q) ^ (Q -> P) T T T T T T F F T F F T T F F F F T T T

39. Laws of Equivalence in Proposition Idempotent Laws P v P = P P ^ P = P Commutative Laws P v Q = Q v P P ^ Q = Q ^ P Associative Laws (P v Q) v R = P v (Q v R) (P ^ Q) ^ R = P ^ (Q ^ R) 2/2/20XX PRESENTATION TITLE 39

40. Laws of Equivalence in Proposition Distribution Laws P v (Q ^ R) = (P v Q) ^ (P v R) P ^ (Q v R) = (P ^ Q) v (P ^ R) Absorption P -> Q = P -> (P ^ Q) De Morgan’s Law ˜ (P v Q) = ˜P ^ ˜Q ˜ (P ^ Q) = ˜ P v ˜ Q 2/2/20XX PRESENTATION TITLE 40

41. Laws of Equivalence in Proposition Double Negation ˜ (˜ P) = P Exportation (P ^ Q) -> R = P -> (Q -> R) 2/2/20XX PRESENTATION TITLE 41

42. Topic Three CATEGORICAL PROPOSITIONS AND VALIDITY OF DEDUCTIVE ARGUMENTS

43. Categorical Propositions A categorical proposition is a proposition that expresses the relationship between two categories or sets. 2/2/20XX PRESENTATION TITLE 43

44. Four Standard Categorical Propositions All S are P 2/2/20XX PRESENTATION TITLE 44

45. Four Standard Categorical Propositions No S are P 2/2/20XX PRESENTATION TITLE 45

46. Four Standard Categorical Propositions Some S are P 2/2/20XX PRESENTATION TITLE 46 x

47. Four Standard Categorical Propositions Some S are not P 2/2/20XX PRESENTATION TITLE 47 x

48. Validity of Deductive Arguments A syllogism is a deductive argument with two premises and one conclusion. A syllogism consisting only of categorical propositions is a categorical syllogism. 2/2/20XX PRESENTATION TITLE 48

49. Let’s Try! Consider the following argument: 2/2/20XX PRESENTATION TITLE 49 Premise 1 : I enjoyed all my math subjects. Premise 2: I got a passing mark in every subject I enjoyed. Conclusion: I got a passing mark in all my math subjects.

50. Let’s Try! Consider the following argument: 2/2/20XX PRESENTATION TITLE 50 Premise 1 : All integers are real numbers. Premise 2: All prime numbers are integers. Conclusion: All prime numbers are real numbers.

51. Let’s Try! Consider the following argument: 2/2/20XX PRESENTATION TITLE 51 Premise 1 : Some earthquakes causes damages to buildings and bridges. Premise 2: Some earthquakes register a 5.0 magnitude or higher. Conclusion: All earthquakes of 5.0 magnitude or higher cause damages to building and bridges.

52. Topic Four Determining Validity of Arguments using Truth Tables

53. How do we determine the validity of an argument? Using truth tables, one can also determine the validity or invalidity of an argument. An argument whose form is given below is valid if P1 ^ P2 -> C is a tautology. Otherwise the said argument is invalid. We write : P1 P2 .:C 2/2/20XX PRESENTATION TITLE 53

54. Let’s Try! Use a truth table to establish the validity of this argument. 2/2/20XX PRESENTATION TITLE 54 P -> Q P .: Q - Since all the possible results are true, then it is a tautology which means this argument is valid. P Q P -> Q (P->Q) ^ P (P->Q)^P->Q T T T T T T F F F T F T T F T F F T F T

55. Let’s Try! Use a truth table to establish the validity of this argument. 2/2/20XX PRESENTATION TITLE 55 P -> Q ˜P .:˜ Q - Since not all the possible results are true, then it is not a tautology which means this argument is invalid. P Q P->Q ˜P (P->Q)^ ˜P ˜Q (P->Q)^ ˜P->Q T T T F F F T T F F F F T T F T T T T F F F F T T T T T

56. Try it! Use a truth table to establish the validity of this argument. 2/2/20XX PRESENTATION TITLE 56 P -> ˜Q P .: ˜Q

57. Topic Five Direct Proof and Indirect Proof

58. Rules of Inference Inference is the process of deriving a proposition from other proposition. There are nine rules of inference that can be used in constructing proof of validity. 2/2/20XX PRESENTATION TITLE 58

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62. Direct Proof and Indirect Proof Direct proof employs the method of assuming the hypothesis to be true and then logically deducing that the conclusion is true using known facts. Indirect proof whereas hangs upon the idea that assuming the conjecture to be false we lead to a contradiction which in turn leads to the case the conjecture should be true. 2/2/20XX PRESENTATION TITLE 62

63. Direct Proof and Indirect Proof Both employ the rules of inference on constructing a proof of validity, but in indirect proof you will be using the direct proof approach then negate the conclusion and deduce the contradiction 2/2/20XX PRESENTATION TITLE 63