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 __________________ truth value to the statement.
Example:
The statement “Davenport is the capital of Iowa” is false.
The negation of that statement “Davenport 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 _________ 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 ______________ the hypothesis and the
conclusion and negates _____________.
Conditional: If a figure is a square, then it is a rectangle.
Contrapositive: If a figure is not a rectangle, then it is not a square.
Writing the 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.
The truth value of an inverse can ______________________ 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 ____________________________
because they have the ______________ truth value.
Summary of key ideas:
Indirect Reasoning
Indirect reasoning occurs when you consider all possibilities and prove
____________________ 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 ___________________.
How to write an indirect proof:
1. State as an assumption the opposite (negation) of what you want to prove.
2. Show that this assumption leads to a contradiction.
3. Conclude that the assumption must be false and that what you want to prove must be true.
Writing an indirect proof:
If Laylon 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:
1. State as an assumption the opposite (negation) of what you want to prove.
2. Show that this assumption leads to a contradiction.
3. Conclude that the assumption must be false and that what you want to prove must be true.
Writing an indirect proof:
A
Given:∆ABC with 𝐵�𝐶�>AC
Prove: ∠ A≇∠ B
Steps to follow:
1. State as an assumption the opposite (negation) of what
you want to prove.
B
C
2. Show that this assumption leads to a contradiction.
3. Conclude that the assumption must be false and that what you want to prove must be true.