LOGIC Name: Teacher: Pd: Page 1
Table of Contents Day 1 Introduction to Logic HW pages 8-10 Day 2 - Conjunction, Disjunction, Conditionals, and Biconditionals HW pages 16-17 #13-34 all, #35 65(every other odd) Day 3 Truth Tables, Tautologies and Logically Equivalent Statements HW pages 21-22 # s 1, 5, 8, 12, 18, 23 Day 4 Law of Detachment HW pages 27-29 # s 1-4,15, 25-28 all Day 5 Law of Contrapositive, Inverse, Converse; Proofs in Logic HW pages 35-37 # s 9, 11, 15, 29, 32 Day 6 Law of Modus Tollens; Invalid arguments HW pages 43-44 #s 2, 6, 10, 12-16 even, 21 Pages 44-45 #s 2-28, even Day 7 The Chain Rule; Law of Disjunctive Inference HW pages 50 52 # s 2, 8, 13, 14, 20, 35 pages 53-54 # s 2, 9, 12, 13, 22, 27, 28 Day 8 De Morgan s Law and Law of Simplification HW - pages 59 60 # s 1 29 every other odd pages 61-62 # s 1-7 odd, 15-21 odd, 27-29 odd Day 9 Practice with Logic Proofs HW pages 63 65 Day 10 Review Exercises HW pages 66 68 SUMMARY PAGE Page 69 Exam Page 2
Introduction to Logic Logic is the study of reasoning. All reasoning, mathematical or verbal, is based on how we put sentences together. A mathematical sentence is a sentence that states a fact or contains a complete idea. A sentence that can be judged true or false is called a statement. In the study of logic, each statement is designated by a letter (p, q, r, etc.) and is assigned a true value (T for true or F for false.) Two types of mathematical sentences are a) open sentence contains a variable and cannot assign a truth value. Ex. She is at the park. Ex. It is fun. b) closed sentence (statement) a sentence that can be judged to be either true or false. Ex. The degree measure of a right angle is 90 0. Ex. There are eight days in a week. Example/Practice 1: Page 3
Negation Symbol: The negation of a statement always has the opposite truth value of the original statement. It is usually formed by adding the word not to the original statement. Ex. p : There are 31 days in January. (True) ~p : There are not 31 days in January. (False) A statement and its negation have opposite truth values. Example/Practice 2: Write the negations for each of the following true sentences. 1. Today s weather is sunny. 2. Alice is not going to the play. 3. x 2 Example 3: Answers: a. b. a. b. Page 4
EXAMPLE 4: Let p represent: January has 31 days Let q represent: Christmas is in December For each given sentence: a. Write the sentence in symbolic form using the symbols below. b. Tell whether the sentence is true or false. 1. It is not the case that January has 31 days a. b. 2. It is not true that Christmas is not in December a. b. 3. January has 31 days. a. b. Truth Table To study sentences where we wish to consider all truth values that could be assigned, we use a device called a truth table. A truth table is a compact way of listing symbols to show all possible truth values for a set of sentences. Example 5: Fill in all missing symbols. Translating Statements from Words to Symbols Page 5
Example 6: Let p represents The bird has wings Let q represents The bird can fly Write each of the following sentences in symbolic form. 1. The bird has wings and the bird cannot fly. 2. If the bird can fly, then the bird has wings. 3. The bird cannot fly or the bird has wings. 4. The bird can fly if and only if the bird has wings. Challenge Problem Summary/Closure Page 6
Exit Ticket 1. 2. Page 7
DAY 1 Homework) Page 8
Translating Statements From Words To Symbols Page 9
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Conjunctions and Disjunctions SWBAT: UWarm - Up In logic, a is a compound sentence formed by combining two sentences (or facts) using the word p: Wash the dishes. q: Vacuum the house. p ^ q: For a conjunction (and) to be true, facts must be true. p q p ^ q Page 11
In logic, a is a compound sentence formed by combining two sentences (or facts) using the word p: Wash the dishes. q: Vacuum the house. p v q: For a disjunction (or) to be true, facts must be true. p q p V q Page 12
UConditional U: if then In logic, a is a compound statement formed by combining 2 sentences (or facts) using the words Refer to the statement: If I come to school then Ms. Williams will give me an A. Let p: I come to school. Let q: Ms. Williams gives me an A. Based on this condition, fill in the accompanying chart. p q p q T T T F F T F F A is a compound sentence formed by combining the two conditionals: p q and q p. Page 13
Summary of Compound Sentences A conjunction is a compound sentence formed by using the word and to combine two simple sentences. p q p q T T T F F T F F A disjunction is a compound sentence formed by using the word or to combine two simple sentences. p q p q T T T F F T F F A conditional is a compound sentence usually formed by using the words if then to combine two simple sentences. p q p q T T T F F T F F A biconditional is a compound sentence formed by combining the two conditionals p q and q p. p q p q T T T F F T F F Page 14
Challenge Problem: Directions: Complete the truth table p q p q p V q Exit Ticket: Page 15
HW Day 2 Assignment Page 16
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Truth Tables, Tautologies, and Logically Equivalent Statements SWBAT: UWarm - Up In logic, a tautology is a compound statement that is always true, no matter what truth values are assigned to the simple sentences within the compound sentence. Example: p q (p q) Example: (p q) (p q) Page 18
Logically Equivalent Statements When two statements have the same truth values, we say that the statements are logically equivalent. To show that equivalence exists between two statements, we use the biconditional, if and only if: If the result is a tautology then the statements are logically equivalent. Ex.1 Let p represent I study. Let q represent I ll pass the test. The two statements being tested for an equivalence are: If I study, then I ll pass the test. I don t study or I ll pass the test. a) Express each statement in symbolic form. b) Express the biconditional in symbolic form. c) Set up a truth table to see if the statements are logically equivalent. Ex. 2 Let c represent Simon takes chorus. Let s represent Simon takes Spanish. a) Using s and c and proper logic connectives, express each of the following sentences in symbolic form. If Simon takes chorus, then he cannot take Spanish. If Simon takes Spanish then he cannot take chorus. b) Prove that the two statements are logically equivalent, or give a reason why they are not equivalent. Page 19
SUMMARY Page 20
HW Day 3 Tautologies and Logically Equivalent Statements Page 21
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Law of Detachment SWBAT: UWarm - Up UDeductive reasoningu is the process of using logic to draw conclusions from given facts, definitions, and properties. In deductive reasoning, if the given facts are true and you apply the correct logic, then the conclusion must be true. The Law of Detachment is one valid form of deductive reasoning. An argument consists of a series of statements called premises and a final statement called a conclusion. We say that the premises lead to the conclusion, or that the conclusion follows the premises. For example: Premise: UPremise: Conclusion: If I play baseball, then I need a bat. I play baseball. I need a bat. Page 23
Example 1: An auto mechanic knows that if a car has a dead battery, the car will not start. A mechanic begins work on a car and finds the battery is dead. What conclusion will she make? Example 2: If there is lightning, then it is not safe to be out in the open. Marla sees lightning from the soccer field. What conclusion will she make? Example 3: If it is snowing, then the temperature is less than or equal to 32 F. The temperature is 20 F. What conclusion can you make? UModel Problems Example 4: Let r represent It is raining. Let m represent I have to mow the lawn. Given the following premises: If it is raining, then I don t have to mow the lawn. It is raining. a. Using r, m, and proper logic connectives, express the premises of this argument in symbolic form. b. Write a conclusion in symbolic form. c. Translate the conclusion into words. Page 24
Hidden Conditionals Sometimes a conditional does not always use the words if. then or the word implies. Such a statement is called a hidden conditional. Ex. A polygon of three sides is called a triangle. If a polygon has three sides, then the polygon is called a triangle. Ex. What s good for Bill is good for me. If it s good for Bill then it s good for me. Ex. I ll get a job when I graduate. If I graduate, then I ll get a job. Ex. Drink milk to stay healthy. If you drink milk, then you will stay healthy. Example 5: Write a valid conclusion for the given premises or indicate that no conclusion is possible. Premises: If adjacent angles are supplementary, then the angles form a linear pair. Example 6: Write a valid conclusion for the given premises or indicate that no conclusion is possible. Premises: Reading will give a person knowledge. Mrs. Williams is an avid reader. Page 25
SUMMARY Exit Ticket Page 26
Day 4 Law of Detachment HW Page 27
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Inverse, Converse, Contrapositive SWBAT: UWarm - Up The converse is formed by interchanging the hypothesis and the conclusion. The inverse is formed by negating the hypothesis and negating the conclusion. The contrapositive is formed by negating both the hypothesis and conclusion, and then interchanging the resulting negations. Ex. Given the conditional : p q Find the converse: Find the inverse: Find the contrapositive: Ex. s: It is spring m: The month is May Conditional (s m): If it is spring, then the month is May. Inverse ( ): Converse ( ): Contrapositive ( ): The inverse and converse of a given conditional do not always have the same truth value as the given conditional. The conditional and its contrapositive are logically equivalent statements. The inverse and converse of a given conditional are logically equivalent to each other. Complete the truth tables. Page 30
Find the inverse, converse and contrapositive of each of the given statements: 1. t ~w 2. ~m p 3. If you use Tickle deodorant, then you will not have body odor. 4. If a man is honest, then he does not steal. UPractice 1. Given the true statement: If a person is eligible to vote, then that person is a citizen. Which statement must also be true? (1) Kayla is not a citizen; therefore, she is not eligible to vote. (2) Juan is a citizen; therefore, he is eligible to vote. (3) Marie is not eligible to vote; therefore, she is not a citizen. (4) Morgan has never voted; therefore, he is not a citizen. 2. What is the inverse of the statement If Julie works hard, then she succeeds? (1) If Julie succeeds, then she works hard. (2) If Julie does not succeed, then she does not work hard. (3) If Julie works hard, then she does not succeed. (4) If Julie does not work hard, then she does not succeed. 2 2 2 3. What is the converse of the statement "If a b c, then ABC is a right triangle"? 2 2 2 (1) If ABC is a right triangle, then a b c. 2 2 2 (2) a b c if, and only if, ABC is a right triangle. 2 2 2 (3) If ABC is not a right triangle, then a b c. 2 2 2 (4) If a b c, then ABC is not a right triangle. Page 31
U p UII. Law of ContrapositiveU: States that when a conditional premise is true, then the contrapositive of the premise if true. OR A conditional and its contrapositive are logically equivalent. q U or (p q) (~q ~p) ~q ~p Proofs in Logic When we are given a series of premises that are true and we apply laws of reasoning to reach a conclusion that is true, we say that we are proving an argument by means of a formal proof. We use two columns. The first column consists of statements and the second column consists of reasons. Given the premises: If Joanna saves enough money, then she can buy a bike. Joanna cannot buy a bike. Prove the conclusion: Joanna did not save enough money. Statements Reasons Your Turn! 4. Given the premises: If Al does not study, then he will fail. Al did not fail. Prove the conclusion: Al studied. **** Make sure you set up a formal proof! **** Page 32
Proofs in Logic Page 33
Exit Ticket 1. 2. 3. Page 34
Day 5 - HW Inverse, Converse, and Contrapositive Page 35
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Law of Modus Tollens and Invalid Arguments SWBAT: UWarm - Up In Ruritania, when Juliet is no longer the queen, then her son Prince Charles will take the throne as the new king. Let us use these facts to present a third law of reasoning. Let p represent Charles is the prince and let q represent Juliet is the queen Premise (p q): UPremise ( q): Conclusion ( p): If Charles is still the prince, then Juliet is still the queen. Juliet is no longer the queen. Charles is no longer the prince. What two laws does the Law of Modus Tollens combine? Page 38
Ex. 1. Write a valid conclusion for the given set of premises: If I am smart, then I like chemistry. I do not like chemistry. Ex. 2. Write a valid conclusion: ~r ~q Uq Ex. 3. Write a formal proof: Ex 4: Write a formal proof Given: The scale is broken if Sal weighs less than 150 pounds. The scale is not broken. Prove: Sal does not weigh less than 150 pounds. Let b represent: The scale is broken Let w represent: Sal weighs less than 150 pounds. Page 39
Invalid Arguments At times, we may be confronted with an argument in which all of its premises are true but these premises do not always lead to a conclusion that is true. The conclusion could be true or false. Such an argument is called an Uinvalid argument. Example 1: The First Invalid Argument Prove why this is invalid. Example 2: Prove why this is invalid. Page 40
Model Problems In 1-4, if the argument is valid, state the law of reasoning that tells why the conclusion is true. If the argument is invalid, write UInvalid. Page 41
SUMMARY U SUMMARY EXIT TICKET 1. 2. 3. Page 42
Day 6 Law of Modus Tollens/Invalid Arguments HW Page 43
Invalid Arguments Page 44
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The Chain Rule and The Law of Disjunctive Inference SWBAT: UWarm - Up Ex 3: If Phil is not on time, then he will be fired. If Phil is fired, then Trude will get his job. Page 46
Ex 4: Given: a c ~a b ~c Prove: b Ex 5: Given: If I walk, then I don t run I do not collapse if I don t run I collapse If I do not walk, then I am out of breath. Prove: I am out of breath Page 47
V. Law of Disjunctive Inference: In the first four laws of inference presented, one or more of the premises was in the form of a conditional statement p q. In this next inference law, there are no conditional statements in the given premises. As we watch a mystery show on television, we believe that the criminal is either the butler or the strange neighbor. We then discover a clue that rules out the neighbor. We can now infer that the butler did it! Let b represent The butler is the criminal, and let n represent The neighbor is the criminal. In words and in symbols we say: Premise (b n): UPremise ( n): Conclusion (b): The butler is the criminal or the neighbor is the criminal. The neighbor is not the criminal. The butler is the criminal. Page 48
Ex. 2. Write a formal proof: Given: c b a ~b c Prove: a SUMMARY Exit Ticket Page 49
Chain Rule HW Day 7: Chain Rule and Law of Disjunctive Inference Page 50
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Law of Disjunctive Inference Page 53
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Negations, De Morgan s Laws, and Law of Simplification SWBAT: UWarm - Up VI. Law of Double Negation ~(~p) and p are logically equivalent statements. VI. DeMorgan s Law The next two laws of inference were discovered by an English mathematician named De Morgan and they bear his name. These inference rules tell how to negate a conjunction and how to negate a disjunction. 1. Negation of a Conjunction The negation of a conjunction of two statements is logically equivalent to the disjunction of the negation of each. 2. Negation of a Disjunction The negation of a disjunction of two statements is logically equivalent to the conjunction of the negation of each. Ex. Write the statement that is logically equivalent to the given statement: a) ~(h ~d) b) It is not true that I will take Gloria to the dance or I will take Patricia to the dance. c) What is the negation of ~ t m? Page 55
Ex 2: Write a statement that is logically equivalent to: It is not true that I will take Gloria to the dance or I will take Patricia to the dance. Ex 3: Page 56
VIII. The Law of Simplification: states that when a single conjunctive premise is true, it follows that each of the individual conjuncts must be true. p q p p q q IX. The Law of Conjunction: states that when two given premises are true, it follows that the conjunction of these premises is true. p q p q X. The Law of Disjunctive Addition: states that when a single premise is true, it follows that any disjunction that has this premise as one of its disjuncts must also be true. p p q Ex 4: Given: r t r m t k Ex 5: Prove: k Page 57
SUMMARY Exit Ticket 1. 2. If the statements c d and p d are true, which statement must also be true? Page 58
De Morgan s Law HW Day 8: De Morgan s Law, and Law of Simplification Page 59
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Law of Simplification Page 61
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Practice with Logic Proofs Page 63
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Review Exercises Page 66
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SUMMARY PAGE Page 69
HOMEWORK ANSWERS Page 70