5-4 Inverses, Contrapositives and Indirect Reasoning
New Vocabulary:
Negation A negation of a statement has the opposite
truth value of the original statement.
Example:
Truth value: true
Statement: Mitzy is a student.
Negation: Mitzy is not a student. Truth value : false
Example:
Statement: The Measure of Angle ABC < 40 degrees
Negation: The measure of Angle ABC >_ 40 degrees
More vocab:
Inverse: An inverse is when you take a conditional statement
and negate (change the truth value of) both the hypothesis
and the conclusion.
Example:
Conditional: If Rachelle studies for the test, then she will
pass the test.
Inverse: If Rachelle does not study for the test, then she
will not pass the test.
Notice the truth values remained the same. This is not always
The case.
Example: 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 conditional statement is true, but the inverse is false.
More vocab:
Contrapositive: The contrapositive of a conditional
statement does two things. It switches the hypothesis
and the conclusion, and negates them both. In other
words, the contrapositive is the inverse of the converse.
Example:
Conditional: If Kevin eats twenty twinkies, then he will get
an upset stomach.
Contrapositive: If Kevin does not have an upset stomach,
then he did not eat twenty twinkies.
Contrapositives ALWAYS have the same truth value of
the original conditional statement.
Contrapositives continued:
Example: If an angle is a straight angle, then its
measure is 180 degrees.
Contrapositive: If an angle is not 180 degrees,
then it is not a straight angle.
Truth values are the same. This is always true of contrapositive
statements.
Summing this up Symbolically:
Statement
Conditional
Example
Symbolic form
If an angle is bisected,
then it is divided into
two congruent angles.
p -------> q
~
Negation (of p) An angle is not bisected
Inverse
If an angle is not bisected
then it is not divided into
two congruent angles.
Contrapositive If an angle is not divided
p
Read
If p, then q
Not p
~ p ---> ~ q
If not p,
then not q.
~q -----> ~p
If not q,
then not p.
into two congruent angles,
then it has not been bisected.
5-4 Part II
Contradiction: a contradiction consists of a logical
incompatibility between two or more propositions.
Example: 1. Triangle ABC is equiangular.
2. Triangle ABC is scalene.
3. Triangle ABC is acute.
Statement 1 contradicts statement 2.
Indirect Reasoning A type of reasoning in which
All possibilities are considered, and then all but
One are proved false.
Example: Your Mom says,”Adriana called a few
Minutes ago.”
Step 1: You have two friends named Adriana
Step 2: You know one of them is at soccer practice
Step 3: You conclude that the other Adriana must have
Been the caller.
Indirect Proofs These are proofs that use indirect
Reasoning.
Step 1. The first step in any indirect proof is to make an
Assumption that the opposite (negation) of what you are
Trying to prove is correct.
Step 2. Next you need to show that this assumption leads
To a contradiction.
Step 3. Conclude that the assumption must be false, so what
You want to prove must be true.
Example:
Prove that an obtuse triangle cannot contain a right angle.
Step 1. Assume that an obtuse triangle can have a right angle
Step 2. There is a contradiction here because every triangle
Contains 180 degrees. Obtuse triangles have one angle greater
Than 90 degrees, and if you add 90 degrees to this, you have
More than 180 degrees.
Step 3: Conclusion : Since my assumption was false, then
Obtuse triangles cannot contain a right angle.