WELCOME Chapter 2: Introduction to Proof 2.1: Foundations for Proof

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Presentation transcript:

WELCOME Chapter 2: Introduction to Proof 2.1: Foundations for Proof Last Night’s Homework: Chapter 1 Study Guide Tonight’s Homework: 2.1 Handout

Warm-Up What patterns do you notice in the 1st, 3rd, and 5th shapes? What patterns do you notice in the 2nd, 4th, and 6th shapes? Solve each equation: a. −𝟑(𝟏+𝟐𝒙)=𝟏𝟓 b. −(𝒏+𝟐)−𝟔𝒏= −𝟑𝟎

Chapter 2: Introduction to Proof

Chapter 2 Section 1

Inductive Vs. Deductive Reasoning

Consider the reasoning used by Emma and Ricky in # 1. Who used inductive reasoning? Who used deductive reasoning?

Inductive Vs. Deductive Induction: Start with some data and determine what can logically be concluded. Example: This pen is black. Her pen is black. Your pen is black. Therefore all pens are black Deduction: Start with an assumed premise and determine what else can logically be true. Example: All men are smelly. Tom is a man. Therefore Tom is smelly.

Hypothesis “If” Conclusion “Then” Conditional Statement: A statement that has two parts, hypothesis and conclusion If I study for the test, then I will get a better grade Hypothesis “If” Conclusion “Then” Propositional Form (If-then): If the hypothesis, then the conclusion

Identify the Hypothesis and Conclusion If the Folsom Bulldogs play football, then they will win. I am going to see a movie, if I have time after school. The shape is a triangle, if it has three connected sides.

Counterexample Counterexample: An example that proves a statement is false. Example: If an animal swims under water then it must be a fish.

Converse Statement Made by switching the hypothesis and conclusion Original: If I see lightning, then I hear thunder Converse: If I hear thunder, then I see lightning

Conditional and Converse Abbreviations If the sun is out, then the weather is good. c h Conditional If h, then c h → c Converse If c, then h c → h

Inverse and Contrapositive Inverse: Made by negating (making opposite) the hypothesis and conclusion of the Conditional. Example: Contrapositive: Made by negating (making opposite) the hypothesis and conclusion of the Converse

Summary of Logical Statements:

Truth Values The truth value of a conditional statement is whether the statement is true or false

Checking for Understanding If I throw a coin into the fountain, it will sink to the bottom. 1. Write a statement about this scenario using inductive reasoning. 2.Write a statement about this scenario using deductive reasoning.

David sees all of the coins on the bottom of the fountain, so he concludes that the coin he throws into the fountain will also sink. David knows the laws of physics concerning mass and gravity, so he concludes that the coin will sink to the bottom of the fountain.