Mathematical Reasoning 4_Mathematicsinthemodernworld.pptx

Mathematical Reasoning part 4 college lesson from Mathematicsinthemodernworld course class

1. The Nature of Mathematics General Education / MATWRLD Engr. Santiago L. Repatacodo

2. Mathematical Reasoning Part 3

3. Mathematical reasoning is the critical skill that enables us make use of all other mathematical skills. Through mathematical reasoning, we are able to reflect solutions to problems and determine whether or not they make sense. It helps to develop critical thinking and to understand mathematics in a more meaningful way. Mathematical reasoning

4. Inductive reasoning - It is the process of making general conclusions based on specific examples. examples: - Every object that I release from my hand falls to the ground. Therefore, the next object I release from my hand will fall to the ground. Inductive and deductive reasoning

5. Inductive reasoning examples: - Every crow I have ever seen is black. Therefore, all crows are black. - Based on the data, the Earth has revolved around the sun following an elliptical path for millions of years. Therefore, the Earth will continue to revolve around the sun in the same manner next year. Inductive and deductive reasoning

6. Deductive reasoning - It is the process of making specific conclusions based on general principles examples: - All men are mortal. I am a man. Therefore, I am mortal. ( General principle: Modus ponens ) Inductive and deductive reasoning

7. Deductive reasoning examples: - Given two supplementary angles with one of them measuring , the measure of the other angle angle is . ( General principle: Supplementary angles add up to . )   Inductive and deductive reasoning

8. Deductive reasoning examples: - If , then . ( General principle: If a, b, and c are real numbers and a = b, then ac = bc )   Inductive and deductive reasoning

9. Comparing these two approaches further, consider science and mathematics. - Science is the application of inductive reasoning to build knowledge based on observable evidence. - Every statement in science is considered a theory. The only way to prove it is to collect more evidence. - In inductive reasoning, there is always the possibility that the future evidence could prove the statement false. Inductive versus deductive reasoning

10. Comparing these two approaches further, consider science and mathematics. - Mathematics on the other hand, is deductive reasoning applied to relations among patterns, shapes, forms, structures, and even changes. - Deductive reasoning is always valid. - It makes use of undefined terms, formally defined terms, axioms, theorems, and rules of inference. Inductive versus deductive reasoning

11. A theorem is a statement that can be shown to be true. It is formulated by using a sequence of statements that form an argument, called proof. The rules of inference tie together the steps of a proof. Inductive versus deductive reasoning

12. An argument is a collection of propositions where it is claimed that one of the propositions called the conclusion follows from the other propositions called the premises of the argument. Definition of argument

13. An argument in a propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises , and the final proposition is called the conclusion . An argument is valid if the truth of all its premises implies that the conclusion is true Definition of argument

14. Example: - Given the following arguments, identify the premises and the conclusion. 1. Extensive exercise is good for the health. Good health guarantees clear thinking. So, I recommend extensive exercise to my students Definition of argument Conclusion Premises

15. Example: - Given the following arguments, identify the premises and the conclusion. 2. I believe that Ken is the best prospect for the highest position in the company. He is very intelligent and articulate. To this day, he does all his duties conscientiously. I have not heard of anyone complain about him since he gets along very well with his subordinates and colleagues. He has a clear vision of the direction the company should take. He is also well respected in the business community. Definition of argument Premises Conclusion

16. An argument is valid if: - the truth of the premises logically guarantees the truth of the conclusion or - whenever the premises are all true, the conclusion is also true. Definition of a valid argument

17. Argument form: . . . _____   Definition of a valid argument Where: , … = premises = conclusion  

18. Propositional form Where: , … = premises = conclusion   Definition of a valid argument

19. An argument is said to be valid if whenever the premises are all true, the conclusion is also true. Thus, the argument : Definition of a valid argument . . . _____   is valid if and only if the propositional form is a tautology. Otherwise, the argument is invalid.  

20. Rules of inference - These are logical forms that are used to deduce new statements from the statements whose truth that we already know. - These rules allow us to make conclusions from given premises. Proof of validity

21. Rules of inference - They deal with propositions which can be inferred from other propositions. These rules consist of antecedents (premises) and consequents (conclusion). - For easy application to arguments, the rules are expressed in argument form. Proof of validity

22. Rules of inference - The rules of inference specify conclusions which can be drawn from assertions known or assumed to be true. These rules are called by special names and are stated in argument form. Proof of validity

23. 1. Addition (Add)   Rules of inference Illustration: “It is below freezing now. Therefore, it is either below freezing or raining now.” It is below freezing OR it is raining now. It is below freezing now.

24. 2. Simplification ( Simp )   Rules of inference Illustration: “It is below freezing and raining now. Therefore, it is below freezing now.” It is below freezing. It is raining now. It is below freezing now.

25. 3. Conjunction ( Conj )   Rules of inference Illustration: “It is below freezing now. It is raining now. Therefore, it is below freezing and raining now.” It is below freezing now. It is raining now. It is below freezing AND raining now.

26. 4. Modus ponens (MP)   Rules of inference Illustration: “If you have access to the network, then you can change your grade. You have access to the network. Therefore, you can change your grade.” You have access to the network You can change your grade. You have access to the network You can change your grade.

27. 5. Modus tollens (MT) q   Rules of inference Illustration: “If you have access to the network, then you can change your grade. You cannot change your grade. Therefore, you do not have access to the network.” You have access to the network You can change your grade. You CANNOT change your grade. You DO NOT have access to the access.

28. 6. Hypothetical Syllogism (HS)   Rules of inference Illustration: “If it rains today, then we will stay home. If we stay home, then we will watch Netflix. Therefore, if it rains today, then we will watch Netflix.” It rains today. We will stay home. We will stay home. We will watch Netflix. If it rains today, THEN we will watch Netflix.

29. 7. Disjunctive Syllogism (DS)   Rules of inference Illustration: “I go to work or I go to the movies. I do not go to work. Therefore, I go to the movies.” I go to work. I go to the movies. I DO NOT go to work. I go to the movies.

30. 8. Constructive dilemma (CD)   Rules of inference Illustration: “If I study well then I will get good grades, and if I finish school then I will be successful. I study well or I finish school. Therefore, I will get good grades or I will be successful.”

31. 9. Destructive Dilemma (DD)   Rules of inference Illustration: “If I go shopping today then I will go to the party tonight, and if I get hungry then I will cook dinner. I will not go to the party tonight or I will not cook dinner. Therefore, I will not go shopping today or I do not get hungry.”

32. The corresponding propositional forms of these rules of inference can be shown as tautologies. As an example, let us consider the truth table of modus tollens Rules of inference q     Argument form Propositional form

33. The corresponding propositional forms of these rules of inference can be shown as tautologies. As an example, let us consider the truth table of modus tollens : Rules of inference   T T T T T F F T F T F T F F F T T F F T F T T F F T T F F F T F T T T T T T T T T F F T F T F T F F F T T F F T F T T F F T T F F F T F T T T T Tautology

34. At this point, we will accept all other rules of inference without verification. We claim that all these rules are valid arguments. The proof that we employ using the definition of valid arguments is what we call the direct or formal proof of validity . Direct proof

35. In a direct proof, the conclusion is established by logically deducing it from the given premises. The premises are used as the starting point, and the conclusion is reached through a series of logical inferences. Formal proof are very clear, precise, and detailed; making it easy for others to follow and understand the result or conclusion being proved. Formal proofs make it possible for others to check and verify the proof. Direct proof

36. Proof by contrapositive Proof by contradiction Proof by induction Proof by constructive counterexample Proof by reduction Other methods

37. Examples: For each of the following arguments, construct a formal proof of validity. State the justification (rule of inference) for each of the line that is not a premise. 1. _______________   Direct proof 3 & 5 - Conj 3 - Add 2 & 4 - MP 1 - Simp

38. Examples: 2. Show that the following is a valid argument. = John is optimistic = John is busy = John will play the lottery = John will visit the casino = John is broke 1. John is not optimistic and John is busy 2. If John plays the lottery then John is optimistic 3. If John does not play the lottery, then he will visit the casino 4. If John visits the casino, then he will be broke. Therefore, John is broke.   Direct proof

39. Examples: 2. Show that the following is a valid argument. Direct proof 1. John is not optimistic and John is busy. 2. If John plays the lottery then John is optimistic. 3. If John does not play the lottery, then he will visit the casino. 4. If John visits the casino, then he will be broke. Therefore, John is broke.   ____________ 7.   1 - Simp 2 & 5 - MT 3 & 6 - MP 4 & 7 - MP = John is optimistic = John is busy = John will play the lottery = John will visit the casino = John is broke  

40. Examples: 3. _______________   Direct proof 1 & 5 - Conj 2 & 4 - MP 3 & 4 - DS 6 & 7 - CD

41. Examples: 4. _______________   Direct proof 5 & 3 - Conj 1 & 2 - HS 6 & 4 - CD

42. Examples: 5. _______________   Direct proof 2 & 4 - CD 5 & 3 - DS 1 - Simp

43. The invalidity of an argument may be verified by showing that its propositional form is not a tautology. Since the propositional form of an argument is an implication, then we should be able to show an instance when the premise is true but the conclusion is false . Proof of invalidity of an argument

44. Since the propositional form of an argument is an implication, then we should be able to show an instance when the premise is true but the conclusion is false . * An implication is false when the premise is true but the conclusion is false. Proof of invalidity of an argument   T F

45. We do not have to construct the whole truth table for the propositional form to do this. All we have to do is find the combination of values that makes the propositional form of the argument false * An implication is false when the premise is true but the conclusion is false. Proof of invalidity of an argument   T F

46. The simplified process of constructing a truth table is called the shortened truth table method of showing the invalidity of argument. One counterexample which gives the truth values of the propositions in the argument that make all the premises true and the conclusion false is enough to disprove the validity of the argument. Proof of invalidity of an argument   T F T T T F

47. Examples: Prove the invalidity of the following argument. 1. _______   Proof of invalidity of an argument  transform to propositional form T F F   F   F     F   F   T   T   T T T F F  

48. Examples: Use the shortened truth table to prove the invalidity of the following argument. 2. ) ___________   Proof of invalidity of an argument F   T   T   F   F   F   F   F   T     T F T T T F

49. Nocon, R. & Nocon E. (2018). Mathematics for the modern world. Aufmann , R., Lockwood, J., Nation, R., Clegg, D., Epp, S., Abad Jr., E. (2018). Mathematics in the modern world (Philippine edition). Rex Book Store, Inc. Rosen, K. (2012). Discrete mathematics and its applications (7 th Edition). McGraw Hill. Johnsonbaugh , R. (2005). Discrete mathematics (6 th International Edition). Pearson Education, Inc. References

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51. Activity 1.3.2 - deadline: February 26 th (Saturday) Quiz 2 Mathematical Reasoning - March 1 st (Thursday) - during your class period only Self-Assessment 1: Nature of Mathematics - available from March 1 st to March 3 rd Mini-Project 1 – Valid and Invalid Arguments - deadline March 8 th (Tuesday) Next live meeting will be on March 3 rd (Thursday) Reminders: