5-4: Inverses, Contrapositives, and Indirect Reasoning

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Presentation transcript:

5-4: Inverses, Contrapositives, and Indirect Reasoning Using elimination of all false statements to prove a statement true

Negation The negation of a statement has the opposite truth value to the statement. Example: The statement “Cedar Rapids is the capital of Iowa” is false. The negation of that statement “Cedar Rapids is not the capital of Iowa” is true.

Writing the Negation For each statement, write the negation: Today is Tuesday. Ms. Vargas went to Wisconsin over break. ∠𝐴𝐵𝐶 is not obtuse. Lines m and n are perpendicular.

Inverse and Contrapositive The inverse of a conditional statement negates both the hypothesis and the conclusion. Example: Conditional: If a figure is a square, then it is a rectangle. Inverse: If a figure is not a square, then it is not a rectangle. The contrapositive of a conditional statement switches the hypothesis and the conclusion and negates both. Contrapositive: If a figure is not a rectangle, then it is not a square.

Inverse and Contrapositive Write the inverse and the contrapositive of the following conditional statement: “If you don’t stand for something, you’ll fall for anything.” – Maya Angelou Inverse: Contrapositive:

Truth value Look back at the examples of the inverse and contrapositive. What is the truth value of an inverse, compared to its conditional statement? What is the truth value of a contrapositive, compared to its conditional statement?

Truth Value The truth value of an inverse is ______________________ that of its conditional statement. The truth value of a contrapositive is ______________________ that of its conditional statement. Contrapositives and Conditional statements are said to be equivalent statements because they have the same truth value.

Summary of key ideas: Statement Example Symbolic Form You Read It Conditional If an angle is a straight angle, then its measure is 180. p  q If p, then q. Negation (of p) An angle is not a straight angle ~p Not p. Inverse If an angle is not a straight angle, then its measure is not 180. ~p~q If not p, then not q. Contrapositive If an angle’s measure is not 180, then it is not a straight angle. ~q~p If not q, then not p.

Indirect Reasoning Indirect reasoning occurs when you consider all possibilities and prove all but one false. The one possibility remaining must then be true. Example: You may have used this kind of reasoning on your exam for multiple choice questions – you could perhaps have ruled out all three possibilities that weren’t true, and then decided the fourth choice must be correct. We can use indirect reasoning to create proofs. This type of proof is an indirect proof.

How to write an indirect proof: State as an assumption the opposite (negation) of what you want to prove. Show that this assumption leads to a contradiction. Conclude that the assumption must be false and that what you want to prove must be true.

Writing an indirect proof: If Chirag spends more than $50 to buy two clothing items, then at least one of the items costs more than $25. Given: The cost of the two items is more than $50. Prove: At least one of the items costs more than $25. Steps to follow: State as an assumption the opposite (negation) of what you want to prove. Show that this assumption leads to a contradiction. Conclude that the assumption must be false and that what you want to prove must be true.

Writing an indirect proof: If Chirag spends more than $50 to buy two clothing items, then at least one of the items costs more than $25. Given: The cost of the two items is more than $50. Prove: At least one of the items costs more than $25. Steps to follow: State as an assumption the opposite (negation) of what you want to prove.

Writing an indirect proof: If Chirag spends more than $50 to buy two clothing items, then at least one of the items costs more than $25. Given: The cost of the two items is more than $50. Prove: At least one of the items costs more than $25. Steps to follow: Show that this assumption leads to a contradiction.

Writing an indirect proof: If Chirag spends more than $50 to buy two clothing items, then at least one of the items costs more than $25. Given: The cost of the two items is more than $50. Prove: At least one of the items costs more than $25. Steps to follow: Conclude that the assumption must be false and that what you want to prove must be true.

Writing an indirect proof: Given:∆𝐴𝐵𝐶 with 𝐵𝐶>𝐴𝐶 Prove: ∠𝐴≇∠𝐵 Steps to follow: State as an assumption the opposite (negation) of what you want to prove. Show that this assumption leads to a contradiction. Conclude that the assumption must be false and that what you want to prove must be true. A B C

Ticket out of class Please write your name on your paper.

Ticket out of class Write the negation of “ABCD is not a convex polygon.” Write the inverse and contrapositive of the conditional statement, “If ∆𝐴𝐵𝐶 is equilateral, then it is isosceles.” Write the first step only of an indirect proof. Prove: A triangle cannot contain two right angles.