LOGIC in general mathematics123456789011

Gen math logic

1. Activity: Intelligence Test The following short IQ test consists of 4 short questions which test your intelligence, and the results will tell you, whether you are truly a manager/leader or a child. The questions like: "How do you put a giraffe into a refrigerator?" are easy — the answers may be not: The questions are NOT that difficult. Get a piece of paper and answer the following questions consecutively.

2. How do you put a giraffe into a refrigerator?

3. How do you put an elephant into a refrigerator?

4. The Lion King is hosting an animal conference. All the animals attend except one. Which animal did not attend?

5. There is a river you must cross but it is used by crocodiles, and you do not have a boat. How do you manage it?

6. LOGIC What is a proposition? Types of Proposition Operations on Proposition Truth Values and Truth Tables Forms of Conditional Proposition Validity of an Argument

7. PROPOSITION

8. What is a proposition? Proposition is a complete declarative sentence that is either true or false, but not both.

9. What is a proposition? Examples: 1. Manila is the capital of the Philippines. 2. La Union is in Pangasinan . 3.   4.  

10. Types of proposition

11. Types of proposition Simple proposition is a complete sentence that conveys one thought with no connecting words. Examples: 1. Five is a counting number. 2. Today is Monday.

12. Types of proposition Compound proposition is a proposition that is being built up by combining propositions using propositional connectives. Examples: 1. Bonnie is early and Clyde was late. 2. Either he took my coat or someone stole it.

13. Types of proposition Propositional connective is an operation that combines two propositions and to yield a new proposition whose truth value depends only on the truth values of the two original propositions.  

14. Types of proposition The following are the propositional connectives used in constructing compound propositions: Symbol Conjunction and Disjunction or Conditional if Biconditional if and only if Negation not Symbol Conjunction and Disjunction or Conditional if Biconditional if and only if Negation not

15. Operations on proposition

16. Operations on proposition Conjunction ( )   It is a connection of two simple statement using the word and or but . Examples: 1. Bonnie is early and Clyde was late. 2. Bonnie is early but Clyde was late.

17. Operations on proposition Disjunction ( )   It is a connection of two simple statement using the word or . Examples: 1. Either he took my coat or someone stole it. 2. I will pass history or I will be sad.

18. Operations on proposition Inclusive Disjunction ( )   It is a connection of two simple statement using the word or but in the inclusive sense of and/or . Examples: 1. The weather forecast calls for rain or snow. 2. I will pass history or I will be happy.

19. Operations on proposition Exclusive Disjunction ( )   It is a connection of two simple statement using the word or but in the exclusive sense of either but not both . Examples: 1. I will get an A or B for this course. 2. I will pass history or I will fail.

20. Operations on proposition Conditional ( )   It is a connection of two simple statement using the word if and then . Examples: If you will get high scores, then you will pass. 2. If you will pass the exam, then you will get a high grade. antecedent antecedent consequence consequence

21. Operations on proposition Biconditional ( )   It is a connection of two simple statement using the word if and only if . Examples: 1. You will get high scores if and only if you will review. 2. You will pass if and only if you will get high scores.

22. Operations on proposition Negation ( )   It is a connection of two simple statement using the word not . Examples: 1. Today is not Monday. 2. I will not pass the subject.

23. Operations on proposition Let p = Today is Monday. q = I am tired. Direction : Transform the following into symbols. STATEMENT SYMBOL 1. Today is not Monday. 2. Today is Monday and I am tired. 3. Today is Monday and I am not tired. 4. Today is not Monday and I am tired. 5. Today is not Monday and I am not tired. 6. Today is Monday or I am tired. 7. Today is not Monday or I am tired.              

24. Operations on proposition Let p = I will go swimming. q = I will go cycling. r = I will go to the movies. Direction : Transform the following into symbols. STATEMENT SYMBOL 1. I will go swimming or I will go cycling, and I will go to the movies. 2. I will go swimming, or I will go cycling and I will go to the movies.    

25. Operations on proposition DOMINANT ORDER OF CONNECTIVES 1. Biconditional 2. Conditional 3. Conjunction/Disjunction 4. Negation DOMINANT ORDER OF CONNECTIVES 1. Biconditional 2. Conditional 3. Conjunction/Disjunction 4. Negation A symbol outside the parentheses dominates or outranks any symbol inside the parentheses.

26. Truth values and truth tables

27. Truth values and truth tables The truth table gives us the truth value of a compound proposition for each possible combination of the truth or falsity of the simple proposition within the compound proposition. If there are n simple proposition in a compound proposition, then there are possible true – false combinations.   If a proposition is true, its truth value is true, denoted by T . If a proposition is false, its truth value is false, denoted by F .

28. Truth values and truth tables T T T T F T T T F F T T F F F T F T T T F F F F F F T T T T T T F T T T F F T T F F F T F T T T F F F F F F T T T F F T T F F T

29. Truth values and truth tables Direction : Perform the indicated operations using a truth table. 1.   2.  

30. Truth values and truth tables A compound proposition that is always true, no matter what the truth values of the propositions that occur in it is called a tautology . A compound proposition that is always false is called a contradiction . A compound proposition that is neither a tautology nor a contradiction is called a contingency .

31. Truth values and truth tables Direction : Perform the operations and determine whether the propositions are tautology, contradiction or contingency. 1.   2.   3.  

32. Logical equivalences LOGICAL EQUIVALENCE NAME Identity Laws Domination Laws Idempotent Laws Double Negation Law Commutative Laws Associative Laws Distributive Laws De Morgan’s Laws LOGICAL EQUIVALENCE NAME Identity Laws Domination Laws Idempotent Laws Double Negation Law Commutative Laws Associative Laws Distributive Laws De Morgan’s Laws

33. FORMS OF CONDITIONAL PROPOSITION

34. Forms of conditional proposition STATEMENT SYMBOLS Conditional Converse Inverse Contrapositive STATEMENT SYMBOLS Conditional Converse Inverse Contrapositive

35. Forms of conditional proposition Direction : Transform the given conditional proposition. Conditional Converse Inverse Contrapositive Let p = A figure is a triangle. q = It is a polygon. If a figure is a triangle, then it is a polygon. If a figure is a polygon, then it is a triangle. If a figure is not a triangle, then it is not a polygon. If a figure is not a polygon, then it is not a triangle.

36. Validity of an argument

37. Validity of an argument An argument is formed when we try to connect bits of evidence ( premises ) in a way that will force the audience to draw a desired conclusion . Examples: The person who robbed the Mini-Mart drives a 1989 Toyota Tercel . Gomez drives a 1989 Toyota Tercel . Therefore, Gomez robbed the Mini – Mart. PREMISES : The person who robbed the Mini-Mart drives a 1989 Toyota Tercel Gomez drives a 1989 Toyota Tercel . CONCLUSION : Therefore, Gomez robbed the Mini-Mart.

38. Validity of an argument Examples: 2. The person who drank my coffee left this fingerprint on the cup. Gomez is the only person in the world who has this fingerprint. Therefore, Gomez is the person who drank my coffee. PREMISES : The person who drank my coffee left this fingerprint on the cup. Gomez is the only person in the world who has this fingerprint. CONCLUSION : Therefore, Gomez is the person who drank my coffee.

39. Validity of an argument Syllogism is a formal argument that is formed by two statements and a conclusion which must be true if the two statements are true. An argument is said to be valid if it is logically impossible to reject the conclusion but accept all of the evidence. An argument is said to be invalid if it is logically possible for the conclusion to be false even though every proposition is assumed to be true.

40. Validity of an argument RULES OF INFERENCE SYMBOLS Contrapositive Reasoning (Modus Tollens ) Direct Reasoning (Modus Ponens) Disjunctive Syllogism Transitive Syllogism Rule of Simplification Rule of Addition Rule of Conjunction RULES OF INFERENCE SYMBOLS Contrapositive Reasoning (Modus Tollens ) Direct Reasoning (Modus Ponens) Disjunctive Syllogism Transitive Syllogism Rule of Simplification Rule of Addition Rule of Conjunction FALLACIES SYMBOLS Fallacy of the Converse Fallacy of the Inverse Affirming the Disjunct Fallacy of the Consequent Denying a Conjunct Improper Transposition FALLACIES SYMBOLS Fallacy of the Converse Fallacy of the Inverse Affirming the Disjunct Fallacy of the Consequent Denying a Conjunct Improper Transposition

41. Validity of an argument Direction : Determine whether the following arguments are valid. 1. If the apartment is damaged, then the deposit won't be refunded. The apartment isn't damaged. Therefore, the deposit will be refunded. 2. If I had a hammer, I would fix the chair I did not fix the chair. Therefore, I have a hammer. 3. All bulldogs are mean-looking dogs. All mean-looking dogs are good watchdogs. Therefore, all bulldogs are good watchdogs.

42. Direction : Determine whether the following arguments are valid. 4.   5.   6.   7.   8.   9.   10.   Validity of an argument