3.1 Conditional Statements,
Converses, Inverses,
Contrapositives
LG.1.G.1 Define, compare and contrast inductive reasoning and deductive
reasoning for making predictions based on real world situations, Venn
diagrams, matrix logic, conditional statements (statement, inverse, converse,
and contrapositive), figural patterns
LG.1.G.6 Give justification for conclusions reached by deductive reasoning
Conditional Statements
If . . . Then

Conditional Statement – A
compound statement formed by
combining two sentences (or facts)
using the words “if…then.”
Example: If you pass all of your
tests, then you will pass the class.
Definitions

Hypothesis: The part of the
sentence that follows the word “IF”

Conclusion: The part of the sentence
that follows the word “THEN”
EXAMPLE: If you do your homework
on time, then you will have a better
grade.
What is the hypothesis?
you do your homework on time
What is the conclusion?
you will have a better grade
It may be necessary to rewrite a
sentence so that it is in conditional
form (“if” first, “then” second).
Example:
All surfers like big waves.
Rewrite as a conditional statement.
If you are a surfer, then you like
big waves.
Truth Value

When you determine whether a
conditional statement is true or false,
you determine its truth value.
Counterexamples

Counterexample:
An example that proves a statement
false
Write a counterexample for the
following conditional statement:
If a student likes math, then he likes
chemistry.
Converse

Converse of a Conditional Statement:
Formed by interchanging the hypothesis
and conclusion of the original statement.
Example:
Conditional:
If the space shuttle was launched, then
a cloud of smoke was seen.
Converse:
If a cloud of smoke was seen, then the
space shuttle was launched.
HINT
Try to associate the logical CONVERSE
with Converse sneakers – think of the
two parts of the sentence “putting on
their sneakers and running to their
new positions.”
Inverse

Inverse of a Conditional
Statement: formed by negating the
hypothesis and negating the
conclusion of the original statement.

Put “not” into the hypothesis and the
conclusion
Example

Conditional: If you grew up in Alaska, then you
have seen snow.

Inverse:
If you did not grow up in Alaska, then you have
not seen snow.
HINT: Hint: To create an INverse, you need to
INsert the word not into both the hypothesis and
the conclusion.
Does the truth value of an inverse have to be
the same as the truth value of the
original conditional statement?
Contrapositive
Contrapositive of a Conditional
Statement:
 formed by negating both the
hypothesis and the conclusion and
then interchanging the resulting
negations.

Put “not” into the hypothesis and the
conclusion, then switch the order.
Example

Conditional:
If 8 is an even number, then 8 is
divisible by 2.

Contrapositive:
If 8 is not divisible by two, then 8
is not an even number.
HINT

For contrapositive, combine both
converse and inverse.

The truth value of a contrapositive is
___________________
the original
The same as
conditional statement.
Conditional, Converse,
Inverse, and Contrapositive
For each statement, write the (a) converse, (b) inverse, and (c) the
contrapositive. Give the truth value for each statement.
1. If a student is on the University of Arkansas
a
b
c
football team, then he is called a Razorback .
TRUE
If a student is called a Razorback, then he is on
the University of Arkansas football team. FALSE
If a student is not on the University of Arkansas
football team, then he is not called a Razorback.
FALSE
If a student is not called a Razorback, then he is
not on the University of Arkansas football team.
TRUE
Conditional, Converse,
Inverse, and Contrapositive
For each statement, write the (a) converse, (b) inverse, and (c) the
contrapositive. Give the truth value for each statement.
2. If a person skateboards well, then they have a
good sense of balance. TRUE
a
b
c
If a person has a good sense of balance, then
they skateboard well. FALSE
If a person does not skateboard well, then they
do not have a good sense of balance. FALSE
If a person does not have a good sense of
balance, then they do not skateboard well.
TRUE
Conditional, Converse,
Inverse, and Contrapositive
For each statement, write the (a) converse, (b) inverse, and (c) the
contrapositive. Give the truth value for each statement.
1.
a
If it thunderstorms, then our pond overflows.
TRUE
If our pond overflows, then it has
thunderstormed. FALSE
b
If it does not thunderstorm, then our pond will
not overflow. FALSE
c
If our pond does not overflow, then it has not
thunderstormed. TRUE
Venn Diagrams

A Venn diagram is a drawing used
to represent a set of numbers or
conditions. Venn diagrams can be
useful in explaining conditional
statements.
Example
If you live in Bentonville, then you are
a Tiger fan.
Hypothesis
Conculsion
you live in
Bentonville
you are a Tiger fan
Write the conditional statement
from the Venn diagram.
Equilateral
Squares
Triangles
Triangles with
3 acute angles
Quadrilaterals
If an object is a
square, then it is a
quadrilateral.
Parallelogram with
perpendicular diagonals
Rhombus
If a triangle is
equilateral, then it
has three acute
angles.
If a parallelogram has
perpendicular
diagonals, then it is a
rhombus.