Chapter 2 Reasoning in Geometry 2.2 Introduction to Logic

Introduction In chapter 2 section 2, we will discuss how we use logic to develop mathematical proofs. When writing proofs, It is important to use exact and correct mathematical language. We must say what we mean!

Introduction Do you recognize the following conversation?

"Then you should say what you mean." the March Hare went on. "I do," Alice hastily replied; "at least -- at least I mean what I say -- that's the same thing, you know. " "Not the same thing a bit!" said the Hatter, "Why, you might just as well say that 'I see what I eat' is the same thing as 'I eat what I see'!"

"You might just as well say," added the March Hare, "that 'I like what I get' is the same thing as 'I get what I like'!“ "You might just as well say," added the Dormouse, who seemed to be talking in his sleep, "that 'I breathe when I sleep' is the same thing as 'I sleep when I breathe'!“ "It is the same thing with you," said the Hatter, and here the conversation dropped, and the party sat silent for a minute.

Charles Dodgson Charles Dodgson lived from 1832 to 1898 Dodgson was a mathematics lecturer and author of mathematics books who is better known by the pseudonym Lewis Carroll. He is known especially for Alice's Adventures in Wonderland.

Conditional Statements In order to analyze statements, we will translate them into a logic statement called a conditional statement. (You will be taking notes now)

Essential Question: How do I recognize and analyze a conditional statement?

10/29/2015Free PowerPoint Template from Definition Hypothesis The if part of a conditional statement

10/29/2015Free PowerPoint Template from Defintion Conclusion The then part of a conditional statement

10/29/2015Free PowerPoint Template from Definition Conditional IF something, THEN something else If a car is a Corvette, then it is a Chevy If you are in this room right now, then you are in Geometry

Conditional Statements  A _________________ is a statement that can be expressed in ________form. conditional statement “if-then” 2. A conditional statement has _________. The __________ is the ____ part. The __________ is the ______ part. hypothesis two parts “if” conclusion“then”

Conditional Statements Example: (Original) I breathe when I sleep (Conditional) If I am sleeping, then I am breathing.

Lesson 2-1 Conditional Statements14 Conditional Statements Definition:A conditional statement is a statement that can be written in if-then form. “If _____________, then ______________.” Example:If your feet smell and your nose runs, then you're built upside down. Continued……

10/29/2015Free PowerPoint Template from Definition Conditional If / then statements are conditional. The then part of the statement is depends on (is conditional to) the if part. In shorthand, the statement is “if p then q” In symbol form, p = feet smell, nose runs q = built upside down

10/29/2015Geometry16 Rewrite in the if-then form All mammals breathe oxygen –If an animal is a mammal, then it breathes oxygen. A number divisible by 9 is also divisible by 3 –If a number s divisible by 9, then it is divisible by 3.

10/29/2015Geometry17 Examples If you are 13 years old, then you are a teenager. Hypothesis: –You are 13 years old Conclusion: –You are a teenager

10/29/2015Free PowerPoint Template from If a car is a Corvette, then it is a Chevrolet Hypothesis Conclusion

10/29/2015Free PowerPoint Template from Euler Diagram (Venn Diagram) Cars Chevys Corvettes

Euler Diagram (Venn Diagram) If a car is a Corvette, then it is a Chevrolet Chevrolets Corvettes (Conclusion: then part) (Hypothesis: If part)

Example: Euler Diagram What is the conditional statement? If two angles form a linear pair, then the angles are supplementary angles Supplementary angles Linear pairs (Conclusion: then part) (Hypothesis: If part)

Conditional Statements The ________ of a conditional statement is formed by switching the hypothesis and the conclusion. Example: converse (Conditional)If I am sleeping, then I am breathing. (Converse) If I am breathing, then I am sleeping.

10/29/2015Free PowerPoint Template from Definition Converse Changing the if and the then around Conditional: If a car is a Corvette, then it is a Chevrolet Converse: If a car is a Chevrolet, then it is a Corvette

10/29/2015Free PowerPoint Template from Determine the Converse If you are wearing a skirt, then you are a female If you are a female, then you are wearing a skirt

10/29/2015Free PowerPoint Template from Definition Counterexample An example that proves a statement false Consider the conditional statement: If you are a female, then you are wearing a skirt Is there any females in the room that are not wearing a skirt?

10/29/2015Geometry26 Writing a Counterexample Write a counterexample to show that the following conditional statement is false –If x 2 = 16, then x = 4. –As a counterexample, let x = -4. The hypothesis is true, but the conclusion is false. Therefore the conditional statement is false.

10/29/2015Free PowerPoint Template from Definition Deductive Reasoning The process of drawing logically certain conclusions by using an argument

Euler Diagram (Venn Diagram) Susan’s car is a Corvette 1.If a car is a Corvette, then it is a Chevrolet 2. Susan’s car is a Corvette 3.Therefore the conclusion is: Susan's car is a Chevrolet. Chevrolets Corvettes Susan’s car

10/29/2015Free PowerPoint Template from Definition If-Then Transitive Property If A then B If B then C You can conclude: If A then C Also known as a logic chain

10/29/2015Free PowerPoint Template from Example Consider the following conditionals - If cats freak, then mice frisk –If sirens shriek, then dogs howl –If dogs howl, then cats freak Prove the following: If sirens shriek, then mice frisk

10/29/2015Free PowerPoint Template from If cats freak, then mice frisk If sirens shriek, then dogs howl If dogs howl, then cats freak First, find the hypothesis of the conditional you are trying to prove Using the provided statements to prove the following conclusion: If sirens shriek, then mice frisk If sirens shriek, then dogs howl Second, write down the conditional with that hypothesis Look for the conditional that begins with the then statement and write it down under the first If dogs howl, then cats freak Keep repeating until you get a conclusion that matches the one you’re looking for If cats freak, then mice frisk Conclusion: If sirens shriek, then mice frisk Logical Chain (Transitive property)

1. Identify the underlined portion of the conditional statement.  hypothesis  Conclusion  neither

2. Identify the underlined portion of the conditional statement.  hypothesis  Conclusion  neither

4. Identify the converse for the given conditional.  If you do not like tennis, then you do not play on the tennis team.  If you play on the tennis team, then you like tennis.  If you do not play on the tennis team, then you do not like tennis.  You play tennis only if you like tennis.

10/29/2015Free PowerPoint Template from Assignment Read pages 90-93, Ch2 Sec 2 Complete problems on Page 95 #9-34 Due Friday Oct. 15. This is an involved set of problems and will take some time to complete. You will be making a big mistake if you wait until Thursday evening to begin this assignment. Suggestion: Break into small parts, complete 6 to 10 problems per day/night.