HS Pre-Algebra Notes Unit 14: Logic. Conditional Statements

Size: px

Start display at page:

Download "HS Pre-Algebra Notes Unit 14: Logic. Conditional Statements"

Dina Wood
6 years ago
Views:

1 HS Pre-Algebra Notes Unit 14: Logic Conditional Statements Objectives: (5.9) The student will identify the hyothesis and conclusion of conditional statements. (5.10) The student will write conditional statements. We often make if-then statements in our everyday conversation. For instance, If I study for the test, then I will get a good grade or If it is raining, then I will get wet are examles of if-then statements. In math, an if-then statement is called a conditional statement. The art that follows the if is called the hyothesis, and the art that follows the then is the conclusion. conditional statement hyothesis conclusion The hyothesis is often reresented by the letter. The conclusion is often shown as. Therefore, a conditional statement is often written as, which is read If, then. Let s ractice writing conditional statements. Examle: Let : There is a red sky at night. Let : Weather will be good the next day. : If there is a red sky at night, then weather will be good the next day. Practice: Let : I get a job. Let : I earn money. Let r: I buy concert tickets. Let s: I send all my money. Use the legend above to translate the following conditional statements: 1. If I get a job, then I earn money. 2. r If I earn money, then I buy concert tickets. 3. r If I get a job, then I buy concert tickets. 4. r s If I buy concert tickets, then I send all my money. Let t: It is raining outside. Let u: I will use an umbrella. Let v: I will get wet. If I study for the test, then I will get a good grade. Use the legend to translate the following from English to symbols. 1. If it is raining outside, I will get wet. t v 2. If it is raining outside, I will use an umbrella. t u 3. If I use an umbrella, it is raining outside. u t Sulementary Materials HS Pre-Algebra, Unit 14: Logic Page 1 of 5

2 Symbols of Logic We know that If, then can be written as. Other symbols used are: not is written therefore is written The statements below illustrate translating from statement form to symbolic form. Statement If yesterday was Tuesday, then today is Wednesday. If yesterday was Tuesday, then today is not Thursday. If I do not study, then I will fail the test. Symbolic Form Let : Yesterday was Tuesday. Let : Today is Wednesday. Let : Yesterday was Tuesday. Let : Today is Thursday. Let : I study. Let : I will fail the test. In the following exercises, shorten the statements by using the symbols of logic. 1. If, then not. 2. If not, then not 3. If not r, then. r 4. If s, then. s Translate the symbolic statements into English statements. Let : The number is even. Let : The number is divisible by 2. Let r: The number is odd. 5. If, then. If the number is not divisible by 2, then the number is not even. 6. If r. If the number is divisible by 2, then the number is not odd. 7. If. If the number is divisible by 2, then the number is even. (Please note that not all conditional statements are necessarily true.) 8. r If the number is even, then the number is odd. 9. If the number is divisible by 2, then the number is not even. Sulementary Materials HS Pre-Algebra, Unit 14: Logic Page 2 of 5

3 More Conditional Statements Every conditional statement has three other conditional statements associated with it. If we write the original statement as, then Converse : Inverse: Contraositive: In logic, conditional statements can be either true or false. If a statement is true, is the converse true or false? Inverse? Contraositive? Try the following examle: Statement: If it is a rose, then it is a flower. true Converse: If it is a flower, then it is rose. false Inverse: If it is not a rose, then it is not a flower. false Contraositive: If it is not a flower, then it is not a rose. true For the following statement, write the converse, inverse, and contraositive. Then decide if each is true or false. Statement: If the oint is reresented by ( 3, 2), then the true oint is in uadrant IV. Converse: If the oint is in uadrant IV, then the oint false 3, 2. is reresented by ( ) Inverse: If the oint is not reresented by ( 3, 2), then false the oint is not in uadrant IV. Contraositive: If the oint is not in uadrant IV, then the oint is 3, 2. not reresented by ( ) true You can make the conjecture that if the statement is true, then only the (converse, inverse, contraositive) is always true. (contraositive) Sulementary Materials HS Pre-Algebra, Unit 14: Logic Page 3 of 5

4 Forms of Valid Reasoning Syllabus Objective: (5.11) The student will justify conclusions to logical arguments. In the world of logic, there are many basic forms of valid reasoning. Two of the most common are shown below: We also know the contraositive is true. The arguments below are examles of arguments showing valid reasoning. They are shown in both English and symbolic form. English Argument Symbolic Translation If Darla had the flu, then Darla gave Marcus Let : Darla had the flu. the flu. Darla had the flu. Therefore, Darla gave Let : Darla gave Marcus the flu. Marcus the flu. If 2x + 5 = 11, then x = 3. But x 3. Let : 2x + 5 = 11 Therefore, 2x Let : x = 3 If two angles are vertical angles, then the two Let : Two angles are vertical angles. angles are congruent. Therefore, if two angles are not Let : The two angles are congruent. congruent, then they are not vertical angles. Now let s ractice. Sulementary Materials HS Pre-Algebra, Unit 14: Logic Page 4 of 5

5 Determine if the following statements do or do not fit the logically valid atterns. Set u a legend to show what each letter reresents; then translate the argument into symbolic form. If it is a valid form of reasoning, identify a conclusion in English. If the statements do not fit a logically valid reasoning attern, write no valid conclusion. Examle: If Marty is a dog, then Marty can bark. Marty cannot bark. Conclusion: Therefore, Marty is not a dog. Let : Marty is a dog. Let : Marty can bark. 1. If students are smart, then students will study. Let : Students are smart. Students are smart. Let : Students will study. Conclusion: Therefore, students will study. 2. If you are oen-minded, then you listen to both Let : You are oen-minded. sides of the story. You do not listen to both sides Let : You listen to both sides of the story. of the story. Conclusion: Therefore, you are not oen-minded. 3. If the triangle is isosceles, then the base angles are Let : The triangle is isosceles. eual. The base angles are not eual. Let : The base angles are eual. Conclusion: Therefore, the triangle is not isosceles. Sulementary Materials HS Pre-Algebra, Unit 14: Logic Page 5 of 5 4. If you buy Snickers candy bars, you like Let : You buy Snickers candy bars. chocolate. You do not buy Snickers. Let : You like chocolate. No valid conclusion 5. If the triangle is a right triangle, then the triangle Let : The triangle is a right triangle. has a 90 angle. The triangle is a right triangle. Let : The triangle has a 90 angle. Conclusion: Therefore, the triangle has a 90 angle. not a valid form

3. Think of things that are made of plastic. Let s see who can name at least three things made of plastic.

3. Think of things that are made of plastic. Let s see who can name at least three things made of plastic. LESSON 75 EXERCISE 1 Actions Verb Tense/Pronouns 1. It s time for some actions. a. Everybody, point to the wall. (Signal. Wait.) What are you doing? (Signal.) Pointing to the wall. Everybody, point to