2.2 Converse, Inverse, and
Contrapositive
Definition

Negation---saying the opposite of what
was being said previously

This can be done by adding “not” to
the statement or by saying “It is not the
case that”
Example #1


Statement:
The dog is big
Negation:
The dog is not big
or
It is not the case that the dog is big
Notation



The symbol we use for negation is ~
We can use a letter to represent a
statement
Ex p: The dog is big
What is ~p:
????
Example #2
Write the negation of the statement in
two ways
p: The football has laces
~p:
or
~p:

Notation

Hypothesis: statement containing the word
“if”
Ex. P: If the instrument is a guitar
Conclusion: Statement containing the word
“then”
Ex. Q: Then the instrument has strings
When putting these together we get p q
Where refers to an if-then relationship

Example #3

Statement:
If p then q =
p
q

Converse:
If q then p =
q
p

Inverse:
If ~p then ~q = ~p
~q

Contrapositive:
If ~q then ~p = ~q
~p
Example #5
Write the inverse of the statement
and tell if it is true or false.
Statement: If two angles are right
angles, then the angles are
congruent p q
Inverse: ~p ~q
Write the inverse and then determine
if it is true or false

Counterexample

Two angles having a
measure of 45 degrees are
congruent but are not right
angles.
Example #5



Write the contrapositive of the statement
and tell whether or not it is true or false
Statement: p q: If a figure is a
rectangle, then the figure is a polygon
Contrapositive ~q ~p:
True or False
Example #6



Write the converse of the statement
and tell whether it is true or false
Statement: If a figure is a rectangle,
then it is a square. True or False
Converse:
True or false
Classwork