Section 3.3
Truth Tables
for the
Conditional
and
Biconditional
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
INB Table of Contents
Date
2.3-2
Topic
Page #
November 18, 2013
Section 3.3 Examples
70
November 18, 2013
Section 3.3 Notes
71
November 18, 2013
Section 3.4 Examples
72
November 18, 2013
Section 3.4 Notes
73
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn

Truth tables for conditional and
biconditional

Self-contradictions

Tautologies

Implications
3.3-3
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Conditional
Case
Case
Case
Case
1
2
3
4
p
T
T
F
F
q
T
F
T
F
p→q
T
F
T
T
The conditional statement p → q is true in
every case except when p is a true statement and
q is a false statement.
3.3-4
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Example: Truth Table with a
Conditional
Construct a truth table for the
statement ~p → ~q.
3.3-5
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Biconditional
The biconditional statement, p ↔ q means that p → q and
q → p or, symbolically (p → q) ⋀ (q → p).
case 1
case 2
case 3
case 4
order of
steps
3.3-7
p
T
T
F
F
q (p 
T T T
F T F
T F T
F F T
1 3
q)
T
F
T
F
2
 (q  p)
T
F
F
T
7
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T
F
T
F
4
T
T
F
T
6
T
T
F
F
5
Biconditional
The biconditional statement, p ↔ q is
true only when p and q have the same
truth value, that is, when both are true
or both are false.
3.3-8
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Example 4: A Truth Table Using
a Biconditional
Construct a truth table for the statement
~p ↔ (~q → r).
3.3-9
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Example 7: Using Real Data in
Compound Statements
The graph represents the student population by age
group in 2009 for the State College of Florida (SCF). Use
this graph to determine the truth value of the following
compound statements.
3.3-11
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Example 7: Using Real Data in
Compound Statements
If 37% of the SCF population is younger than 21 or 26% of the SCF
population is age 21–30, then 13% of the SCF population is age 31–
40.
3.3-13
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Example 7: Using Real Data in
Compound Statements
3% of the SCF population is older than 50 and 8% of the
SCF population is age 41–50, if and only if 19% of the
SCF population is age 21–30.
3.3-16
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Self-Contradiction

A self-contradiction is a compound
statement that is always false.

When every truth value in the answer column
of the truth table is false, then the statement is
a self-contradiction.
3.3-19
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Example 8: All Falses, a SelfContradiction
Construct a truth table for the statement
(p ↔ q) ⋀ (p ↔ ~q).
3.3-20
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Tautology

A tautology is a compound statement that is
always true.

When every truth value in the answer column
of the truth table is true, the statement is a
tautology.
3.3-22
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Example 9: All Trues, a Tautology
Construct a truth table for the statement
(p ⋀ q) → (p ⋁ r).
3.3-23
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Implication

An implication is a conditional statement that
is a tautology.

The consequent will be true whenever the
antecedent is true.
3.3-25
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Example 10: An Implication?
Determine whether the conditional statement
[(p ⋀ q) ⋀ q] → q is an implication.
3.3-26
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Section 3.4
Equivalent
Statements
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn

Equivalent statements

DeMorgan’s Law

Variations of conditional statements
3.4-29
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Equivalent Statements
Two statements are equivalent,
symbolized ⇔ or ≡, if both statements
have exactly the same truth values in
the answer columns of the truth
tables.
3.4-30
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Equivalent Statements

If the answer columns are not
identical, the statements are not
equivalent.

Sometimes the words logically
equivalent are used in place of the
word equivalent.
3.4-31
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Example 1: Equivalent Statements
Determine whether the following two statements
are equivalent.
p ⋀ (q ⋁ r)
3.4-32
(p ⋀ q) ⋁ (p ⋀ r)
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Example 3: Which Statements Are
Logically Equivalent?
Determine which statement is logically equivalent to “it is not true that
the tire is both out of balance and flat.”
a) if the tire is not flat, then the tire is not out of balance.
b) the tire is not out of balance or the tire is not flat.
c) the tire is not flat and the tire is not out of balance.
d) if the tire is not out of balance, then the tire is not flat.
3.4-35
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De Morgan’s Laws
~ (p  q)  ~ p  ~ q
~ (p  q)  ~ p  ~ q
3.4-41
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Example 5: Using De Morgan’s Laws
to Write an Equivalent Statement
Write a statement that is logically equivalent to
“It is not true that tomatoes are poisonous or
eating peppers cures the common cold.”
3.4-42
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The Conditional Statement
Written as a Disjunction
p  q  ~p  q
To change a conditional statement
into a disjunction, negate the
antecedent, change the conditional
symbol to a disjunction symbol, and
keep the consequent the same.
3.4-45
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The Disjunction Written as a
Conditional Statement
~p  q  p  q
To change a disjunction statement to a
conditional statement, negate the first
statement, change the disjunction symbol to a
conditional symbol, and keep the second
statement the same.
3.4-46
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Example 7: Rewriting a Disjunction
as a Conditional Statement
Write a conditional statement that is logically equivalent
to “The Oregon Ducks will win or the Oregon State
Beavers will lose.” Assume that the negation of winning is
losing.
3.4-47
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The Negation of the Conditional
Statement Written as a
Conjunction
~(p → q) ≡ p ⋀ ~q
3.4-50
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Example 9: Write an Equivalent
Statement
Write a statement that is equivalent to “It is false that if
you hang the picture then it will be crooked.”
3.4-51
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Variations of the Conditional
Statement
The variations of conditional statements are the
converse of the conditional, the inverse of the
conditional, and the contrapositive of the
conditional.
3.4-53
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Variations of the Conditional
Statement
Name
Conditional
Converse
Inverse
Symbolic Read
Form
p → q “If p, then q”
q→p
“If q, then p”
~p → ~q “If not p, then not q”
Contrapositive ~q → ~p “If not q, then not p”
3.4-54
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 10: The Converse,
Inverse, and Contrapositive
For the conditional statement “If the song contains sitar
music, then the song was written by George Harrison,”
write the
a) converse.
b) inverse.
c) contrapositive.
3.4-55
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