GEOMETRY. Chapter 2: LOGIC. Name: Teacher: Pd:
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1 GEOMERY Chapter 2: LOGIC Name: eacher: Pd:
2 able of Contents DAY 1: SWBA: Identify, write and analyze the different types of logical statements. Pgs: 2-8 Homework: Pgs 6 8 (EVEN ONLY) DAY 2: SWBA: Write and analyze the truth values of Conjunctions and Disjunctions Pgs: 9-14 Homework: Pgs (EVEN ONLY) DAY 3: SWBA: Identify, write, and analyze the truth value of conditional and biconditional statements. Pgs: Homework: Pgs (EVEN ONLY) DAY 4: ull Period Quiz: Day 1 to DAY 3 Homework: Pgs 6 8 (ODD ONLY) Pgs (ODD ONLY) Pgs (ODD ONLY) DAY 5: SWBA: Identify, write, and analyze the truth value of the Converse, Inverse, and Contrapositive Pgs: Homework: Pgs 25 (ALL) DAY 6: Logic Practice est 1
3 DAY 1 SWBA: Identify, write and analyze the different types of logical statements. Warm Up 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 ( for true or for false.) wo types of mathematical sentences are a) open sentence contains a variable and cannot assign a truth value. Ex. She is at the park. Ex. Basketball is fun. b) closed sentence (statement) a sentence that can be judged to be either true or false. Ex. he degree measure of a right angle is Ex. here are eight days in a week. 2
4 Example 1: Negation Symbol: he 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 : here are 31 days in January. (rue) ~p : here are not 31 days in January. (alse) A statement and its negation have opposite truth values. Example 2: Write the negations for each of the following true sentences. 1. oday s weather is sunny. 2. Alice is not going to the play. 3. x 2 3
5 Example 3: Answers: a. b. b. b. EXAMPLE 4 Let p represent: January has 31 days Let q represent: Christmas is in December or each given sentence: a. Write the sentence in symbolic form using the symbols below. b. ell whether the sentence is true or false. 1. It is not the case that January has 31 days 2. It is not true that Christmas is not in December 3. January has 31 days. ruth able o 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: ill in all missing symbols. 4
6 ranslating Statements from Words to Symbols Example 6: Let p represents he bird has wings Let q represents he bird can fly Write each of the following sentences in symbolic form. 1. he bird has wings and the bird cannot fly. 2. If the bird can fly, then the bird has wings. 3. he bird cannot fly or the bird has wings. 4. he bird can fly if and only if the bird has wings. 5
7 DAY 1 - Homework 6
8 a. b. ranslating Statements rom Words o Symbols 7
9 8
10 DAY 2 SWBA: Write and analyze the truth values of Conjunctions and Disjunctions Warm Up Write each sentence in symbolic form, using the given symbols. Let p represent It is cold. Let q represent It is snowing. Let r represent he sun is shining. 1. It is not cold and it is snowing. 2. It is snowing or the sun is not shining. 3. If it is snowing, then the sun is not shining. 4. It is not the case that it is snowing if and only if it is not cold. One of the goals of studying mathematics is to develop the ability to think critically. he study of critical thinking, or reasoning, is called. A is formed when two or more thoughts are connected in one sentence. Example 1: In logic, a is a compound sentence formed by combining two sentences (or facts) using the word p q p ^ q p: Wash the dishes. q: Vacuum the house. p ^ q: Wash the dishes and vacuum the house. or a conjunction (and) to be true, facts must be true. 9
11 Practice: In 1 4, or each given statement: A. Write the statement in symbolic form, using the symbols shown below. B. ell whether the statement is true or false. Let b represent Water boils at 100 C. Let f represent Water freezes at 0 C. Let t represent Normal body temperature is 37 C. Let r represent Room temperature is 60 C. (rue) (rue) (rue) (alse) 1. Normal body temperature is 37 C and water boils at 100 C. 2. Normal body temperature is 37 C and room temperature is 60 C. 3. Water does not boil at 100 C and water does not freeze at 0 C. 4. Room temperature is not 60 C and water freezes at 0 C. Example 2: In logic, a is a compound sentence formed by combining two sentences (or facts) using the word p q p V q p: Wash the dishes. q: Vacuum the house. p v q: Wash the dishes and vacuum the house. or a disjunction (or) to be true, facts must be true. 10
12 Practice: In 1 4, or each given statement: A. Write the statement in symbolic form, using the symbols shown below. B. ell whether the statement is true or false. Let c represent A meter contains 100 centimeters. (rue) Let m represent A meter contains 1,000 millimeters. (rue) Let k represent A kilometer is 1,000 meters. (rue) Let l represent A meter is a liquid measure. (alse) 1. A meter contains 1,000 millimeters or a kilometer is 1,000 meters. 2. A meter contains 100 centimeters or a meter is a liquid measure. 3. A kilometer is not 1,000 meters or a meter does not contain 100 centimeters. 4. It is not the case that a meter contains 100 centimeters or 1,000 millimeters. 1. r w 2. w z 3. w r 4. (r z) 11
13 Summary A conjunction is a compound sentence formed by using the word and to combine two simple sentences. p q p q A disjunction is a compound sentence formed by using the word or to combine two simple sentences. p q p q Challenge Problem: Directions: Complete the truth table p q p q p V q 12
14 DAY 2 - Homework: Conjunction 13
15 Disjunction 14
16 DAY 3 SWBA: Identify, write, and analyze the truth value of conditional and biconditional statements. Warm Up 2.ind two supplement angles in the ratio of 7:5 In logic, a is a compound statement formed by combining sentences (or facts) using the words p is known as the. q is known as the. 15
17 Conditionals: p: (fact 1) If you come to school. q: (fact 2) Ms. Williams will gives me an A. p q: If you come to school, then Ms. Williams will give me an A. A is when a true hypothesis leads to a conclusion. In all other cases, the conditional is. hink of it as a broken promise. Example 1: 16
18 Example 2: Biconditionals: p q In logic, a is a compound statement formed by combining the conditional and the converse using the word. 17
19 Example 3: In 1-4, for each given statement: a. Write the statement in symbolic form, using the symbols shown below. b. ell whether the statement is true or false. Let p represent It is cold. Let q represent it is snowing. Let r represent he sun is shining. (rue) (alse) (rue) 1. It is cold if and only if it is snowing. 2. he sun is shining if and only if it is snowing. 3. It is not cold if and only if the sun is shining. 4. It is not cold if and only if the sun is not shining. Challenge Problem: 18
20 Summary A conditional is a compound sentence usually formed by using the words if then to combine two simple sentences. p q p q A biconditional is a compound sentence formed by combining the two conditionals p q and q p. p q p q 19
21 DAY 3 - Homework: 20
22 . 21
23 DAY 5 SWBA: Identify, write, and analyze the truth value of the Converse, Inverse, and Contrapositive Warm Up Review: In logic, a conditional is a compound statement formed by combining two sentences (or facts) using the words "if... then." p q p is known as the premise, hypothesis or antecedent. q is known as the conclusion or consequent. he of a conditional statement is formed by the and of the original statement. In other words, the parts of the sentence change places. he words "if" and "then" do not move. Conditional: If you study, then you will get an A Converse: If, then Symbolic orm p: you study. q: you will get an A. p q: If you study, then you will get an A. Example 1: 22
24 Practice: 1. Which statement is the converse of If the sum of two angles is 180, then the angles are supplementary? (1) If two angles are supplementary, then their sum is 180. (2) If the sum of two angles is not 180, then the angles are not supplementary. (3) If two angles are not supplementary, then their sum is not 180. (4) If the sum of two angles is not 180, then the angles are supplementary. 2. What is the converse of the statement "If it is Sunday, then I do not go to school"? (1) If I do not go to school, then it is Sunday. (2) If it is not Sunday, then I do not go to school. (3) If I go to school, then it is not Sunday. (4) If it is not Sunday, then I go to school. he of a conditional statement is formed by the hypothesis and the conclusion of the original statement. In other words, the word "not" is added to both parts of the sentence. Conditional: If you study, then you will get an A Inverse: If, then Symbolic orm An interesting fact: he inverse has the same truth value as the converse of the original statement. he INVERSE and the CONVERSE of the original statement are. ("equivalent" means "the same") Practice: 1. What is the inverse of the statement If Mike did his homework, then he will pass this test? (1) If Mike passes this test, then he did his homework. (2) If Mike does not pass this test, then he did not do his homework. (3) If Mike does not pass this test, then he only did half his homework. (4) If Mike did not do his homework, then he will not pass this test. 2. What is the inverse of the statement If I do not buy a ticket, then I do not go to the concert? (1) If I buy a ticket, then I do not go to the concert. (2) If I buy a ticket, then I go to the concert. (3) If I go to the concert, then I buy a ticket. (4) If I do not go to the concert, then I do not buy a ticket. 23
25 he of a conditional statement is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations. In other words, the negates and switches the parts of the sentence. It does BOH the jobs of the INVERSE and the CONVERSE. Conditional: If you study, then you will get an A Contrapositive: If, then Symbolic orm **If the original statement is RUE, the contrapositive is RUE. If the original statement is ALSE, the contrapositive is ALSE. hey are said to be logically equivalent. ("equivalent" means "the same") Practice: 1. What is the contrapositive of the statement If I study, then I pass the test? (1) I pass the test if I study. (2) If I do not study, then I do not pass the test. (3) If I do not pass the test, then I do not study. (4) If I pass the test, then I study. 2. Which statement is logically equivalent to If it is Saturday, then I am not in school? (1) If I am not in school, then it is Saturday. (2) If it is not Saturday, then I am in school. (3) If I am in school, then it is not Saturday. (4) If it is Saturday, then I am in school. Challenge Problem: Homework: 24
26 25
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