3-3: Converse, Inverse, & Contrapositive
Objectives:
Assignment:
1. To write and
• P. 34: 11-16
determine the truth
• P. 36: 16-33
value of the
• Challenge Problems
converse, inverse,
and contrapositive of
a conditional
2. To write and interpret
biconditional
statements
Objective 1
You will be able to write
and determine the truth
value of the converse,
inverse, and
contrapositive of a
conditional
If I’m a skunk, then I skink!
Exercise 1
What is the opposite of the following
statements?
1. The ball is red.
2. The cat is not black.
Negation
The negation of a statement is the
opposite of the original statement.
Statement:
The sick boy eats meat.
Negation:
The sick boy does
not eat meat.
Notice that only the verb of the sentence
gets negated.
Symbolic Notation
Mathematicians are notoriously lazy, creating
shorthand symbols for everything. Conditional
statements are no different.
Symbol
Concept
𝑝
Original Hypothesis
𝑞
Original Conclusion
→
“Implies”
~
“Not”
𝑝→𝑞
“p implies q” “if p, then q”
~𝑝
“not p”
All Kinds of Conditionals
So the symbols make conditionals easy and fun!
Statement
Symbols
Conditional
𝑝→𝑞
Converse
𝑞→𝑝
Inverse
~𝑝 → ~𝑞
Contrapositive
~𝑞 → ~𝑝
Objective 2
You will be able to write and
interpret biconditional statements
All Kinds of Statements
Here are some examples of writing the
converse, inverse, and contrapositive of a
conditional statement.
Exercise 2
Write the converse, inverse, and
contrapositive of the conditional statement.
Indicate the truth value of each statement.
If a polygon is regular, then it is equilateral.
Which of the statements that you wrote are
equivalent?
Equivalent Statements
When pairs of statements are both true or both
false, they are called equivalent statements.
A conditional and its
contrapositive are
equivalent.
An inverse and the converse
are equivalent.
So if a
conditional is
true, so its
contrapositive.
Definitions in Geometry
In geometry, definitions can be written in ifthen form. It is important that these
definitions are reversible. In other words,
the converse of a definition must also be
true.
If a polygon is a hexagon, then it has exactly six sides.
-ANDIf a polygon has exactly six sides, then it is a hexagon.
Perpendicular Lines
If two lines intersect
to form a right
angle, then they are
perpendicular
lines.
Exercise 3
Write the converse
of the definition
of perpendicular
lines.
If two lines intersect
to form a right
angle, then they are
perpendicular
lines.
Biconditional
A biconditional is a statement that combines a
conditional and its true converse in “if and only if”
form.
If a polygon is a hexagon, then it has exactly six sides.
-ANDIf a polygon has exactly six sides, then it is a hexagon.
A polygon is a hexagon if and only if
it has exactly six sides.
Exercise 4
Write the definition
of perpendicular
lines as a
biconditional
statement.
If two lines intersect
to form a right
angle, then they are
perpendicular
lines.
Exercise 5
The definition of a right angle is as follows:
An angle is a right angle iff it measures 90°.
Write the two statements, the original
conditional and its converse, that make up
this definition.
3-3: Converse, Inverse, & Contrapositive
Objectives:
Assignment:
1. To write and
• P. 34: 11-16
determine the truth
• P. 36: 16-33
value of the
• Challenge Problems
converse, inverse,
and contrapositive of
a conditional
2. To write and interpret
biconditional
statements