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Lesson 5.4
Conditional Statements
pp. 176-181
Objectives:
1. To define and write conditional
statements.
2. To define biconditional statements.
3. To define and symbolize the
inverse, converse, and
contrapositive of a conditional.
Definition
A conditional statement is a
statement of the form “If p, then
q,” where p and q are
statements. The notation for this
conditional statement is pq.
Notice that the p represents
the “if” statement, called the
hypothesis, and the q
represents the “then”
statement, called the
conclusion.
EXAMPLE 1 Write the following
statements in if-then form.
a. There are no clouds in the sky, so
it is not raining.
If there are no clouds in the sky,
then it is not raining.
EXAMPLE 1 Write the following
statements in if-then form.
b. School will be canceled if a
blizzard hits.
If a blizzard hits, then school will
be canceled.
Truth table for a conditional statement
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
EXAMPLE 2 Which is true?
If horses had wings, then horses
could fly.
If ocean water is grade-A milk, then
ocean water is nourishing beverage.
If whales walk, then 4 + 1 = 5.
All three are true, since each hypothesis
is false. The symbolic forms are F  F,
F  F, and F  T.
Definition
A biconditional statement is a
statement of the form “p, if and
only if q,” (symbolized by pq),
which means pq and qp.
Theorem 5.1
The conditional pq is
equivalent to the disjunction
~p or q.
p
T
T
F
F
q pq p pq (pq)(pq)
T T
F
T
T
F F
F
F
T
T T
T
T
T
F T
T
T
T
EXAMPLE 3 Change the
following conditional
statement to a disjunction. If a
child disobeys, then he will be
disciplined.
A child obeys, or the child will be
disciplined.
Definition
The converse of a conditional
statement is obtained by
switching the hypothesis and
conclusion. The converse of pq
is qp.
Definition
The inverse of a conditional
statement is obtained by
negating both the hypothesis and
conclusion. The inverse of pq
is ~p~q.
Definition
The contrapositive of a
conditional statement is obtained
by switching and negating the
hypothesis and conclusion. The
contrapositive of pq is ~q~p.
EXAMPLE 4 Write the converse,
inverse, and contrapositive of the
implication below.
Implication If we have a blizzard,
then school will be
canceled.
Converse If school is canceled,
then we had a
blizzard.
EXAMPLE 4 Write the converse,
inverse, and contrapositive of the
implication below.
Implication If we have a blizzard,
then school will be
canceled.
Inverse If we do not have a
blizzard, then school
will not be canceled.
EXAMPLE 4 Write the converse,
inverse, and contrapositive of the
implication below.
Implication If we have a blizzard,
then school will be
canceled.
Contrapositive If school is not
canceled, then we did
not have a blizzard.
Practice: Give the converse,
inverse, and contrapositive of “if two
angles are congruent, then they have
the same measure.”
Converse:
If two angles have the same
measure, then they are
congruent.
Practice: Give the converse,
inverse, and contrapositive of “if two
angles are congruent, then they have
the same measure.”
Inverse:
If two angles are not
congruent, then they do not
have the same measure.
Practice: Give the converse,
inverse, and contrapositive of “if two
angles are congruent, then they have
the same measure.”
Contrapositive:
If two angles do not have
the same measure, then they
are not congruent.
Theorem 5.2
Contrapositive Rule. A
conditional statement is
equivalent to its
contrapositive. In other words,
pq is equivalent to ~q~p.
25.
(pq)
p q pq ~q ~p ~q~p (~q~p)
TT T F F
T
T
TF F T F
F
T
FT T F T
T
T
FF
T
T T
T
T
Homework
pp. 180-181
►A. Exercises
Write the following statements in if-then
form.
1. When I study my geometry, I get good
grades.
If I study my geometry, then
I get good grades.
►A. Exercises
Write the following statements in if-then
form.
5. The flowers bloom because the sun
shines.
If the sun shines, then the
flowers will bloom.
►A. Exercises
State whether the conditionals are true or
false.
9. If our pig is a clean animal, then we
will keep it in the house.
True
►A. Exercises
State whether the conditionals are true or
false.
11. The roads will be slick if and only if
there is ice on the roads.
False
►B. Exercises
Write each biconditional as two
conditionals.
17. You will get an A in geometry if and
only if you study hard.
If you study hard, then you will
get an A in geometry.
If you get an A in geometry, then
you studied hard.
►B. Exercises
Change each conditional to a disjunction.
21. If the stoplight is green, then you can
go.
The stoplight is not green, or you
can go.
■ Cumulative Review
True/False
30. If a triangle is isosceles, then it is
equilateral.
■ Cumulative Review
True/False
31. Theorem 1.3 could be worded “if two
lines intersect, then there is a plane
containing them; and if two lines
intersect, then there is at most one
plane containing them.”
■ Cumulative Review
How many sides or faces does each figure
have?
32. heptagon
■ Cumulative Review
How many sides or faces does each figure
have?
33. icosahedron
■ Cumulative Review
How many sides or faces does each figure
have?
34. decagon