Chapter 2 Reasoning and Proof: Chapter 2 Reasoning and Proof 2.1 Conditional Statements
Introduction: Introduction In chapter 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: Introduction Do you recognize the following conversation?
Slide 4: "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'!"
Slide 5: "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 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 .
2.1 Conditional Statements: 2.1 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:: Essential Question: How do I recognize and analyze a conditional statement?
2.1 Conditional Statements: 2.1 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”
2.1 Conditional Statements: 2.1 Conditional Statements Example: (Original) I breathe when I sleep ( Conditional ) If I am sleeping, then I am breathing.
2.1 Conditional Statements: 2.1 Conditional Statements To fully analyze this conditional statement, we need to find three new conditionals: Converse Inverse Contrapositive
2.1 Conditional Statements: 2.1 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.
2.1 Conditional Statements: 2.1 Conditional Statements The ________ of a conditional statement is formed by negating (inserting “not”) the hypothesis and the conclusion. Example: inverse (Conditional) If I am sleeping , then I am breathing . (Converse) If I am not sleeping, then I am not breathing.
2.1 Conditional Statements: 2.1 Conditional Statements The ______________ of a conditional statement is formed by negating the hypothesis and the conclusion of the converse. Example: (Converse) If I am breathing, then I am sleeping. (Contrapositive) If I am not breathing, then I am not sleeping. contrapositive
2.1 Conditional Statements: 2.1 Conditional Statements Conditional ( ) Inverse ( ) Converse ( ) Contrapositive ( ) If I am sleeping, then I am breathing . If I am not sleeping, then I am not breathing. If I am breathing, then I am sleeping. If I am not breathing, then I am not sleeping. if…then insert not switch switch and insert not
2.1 Conditional Statements: 2.1 Conditional Statements The conditional statement, inverse, converse and contrapositive all have a truth value. That is, we can determine if they are true or false . When two statements are both true or both false, we say that they are logically equivalent .
2.1 Conditional Statements: 2.1 Conditional Statements Conditional If m <A = 30 ° , then <A is acute. Inverse (insert not) Converse (switch) Contrapositive (switch then insert not) T T F F If m <A ≠ 30 °, then <A is not acute. If <A is acute, then m <A = 30 ° . If <A is not acute, then m <A ≠ 30 ° .
2.1 Conditional Statements: 2.1 Conditional Statements The conditional statement and its contrapositive have the same truth value. They are both true . They are logically equivalent.
2.1 Conditional Statements: 2.1 Conditional Statements The inverse and the converse have the same truth value. They are both false . They are logically equivalent.
Practice: Practice Translate the following statement into a conditional statement. Then find the converse, inverse and contrapositive. “ A cloud of steam can be seen when the space shuttle is launched ”
1. Identify the underlined portion of the conditional statement.: 1. Identify the underlined portion of the conditional statement. hypothesis Conclusion neither
2. Identify the underlined portion of the conditional statement.: 2. Identify the underlined portion of the conditional statement. hypothesis Conclusion neither
3. Identify the underlined portion of the conditional statement.: 3. Identify the underlined portion of the conditional statement. hypothesis Conclusion neither
4. Identify the converse for the given conditional.: 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.
5. Identify the inverse for the given conditional.: 5. Identify the inverse for the given conditional. If 2x is not even, then x is not odd. If 2x is even, then x is odd. If x is even, then 2x is odd. If x is not odd, then 2x is not even.