### The Conditional

```Chapter 3
Introduction
to Logic
© 2008 Pearson Addison-Wesley.
All rights reserved
Chapter 3: Introduction to Logic
3.1
3.2
3.3
3.4
3.5
3.6
Statements and Quantifiers
Truth Tables and Equivalent Statements
The Conditional and Circuits
More on the Conditional
Analyzing Arguments with Euler Diagrams
Analyzing Arguments with Truth Tables
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-2
Chapter 1
Section 3-3
The Conditional and Circuits
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-3
The Conditional and Circuits
• Conditionals
• Negation of a Conditional
• Circuits
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-4
Conditionals
A conditional statement is a compound
statement that uses the connective if…then.
The conditional is written with an arrow,
so “if p then q” is symbolized
p  q.
We read the above as “p implies q” or “if p
then q.” The statement p is the antecedent,
while q is the consequent.
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-5
Truth Table for The Conditional,
If p, then q
If p, then q
pq
p q
T
T
T
T
F
F
F
T
T
F
F
T
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-6
Special Characteristics of Conditional
Statements
1. p  q is false only when the antecedent is
true and the consequent is false.
2. If the antecedent is false, then p  q is
automatically true.
3. If the consequent is true, then p  q is
automatically true.
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-7
Example: Determining Whether a
Conditional Is True or False
Decide whether each statement is True or
False (T represents a true statement, F a false
statement).
a) T  (4  2)
b) (8  1)  F
Solution
a) False
b) True
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-8
Tautology
A statement that is always true, no matter
what the truth values of the components, is
called a tautology. They may be checked
by forming truth tables.
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-9
Negation of a Conditional p  q
The negation of p  q is p 
© 2008 Pearson Addison-Wesley. All rights reserved
q.
3-3-10
Writing a Conditional as an “or”
Statement
p  q is equivalent to
p  q.
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-11
Example: Determining Negations
Determine the negation of each statement.
a) If you ask him, he will come.
b) All dogs love bones.
Solution
a) You ask him and he will not come.
b) It is a dog and it does not love bones.
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-12
Circuits
Logic can be used to design electrical circuits.
p
p
q
Series circuit
q
Parallel circuit
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-13
Equivalent Statements Used to
Simplify Circuits
p  q  r    p  q   p  r 
p  q  r    p  q   p  r 
pq q p
p q  pq
p p  p
 p  q 
 p  q 
p p  p
p
q
p
q
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-14
Equivalent Statements Used to
Simplify Circuits
If T represents any true statement and F
represents any false statement, then
pT T
pF F
p
p
pT
p  F.
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-15
Example: Drawing a Circuit for a
Conditional Statement
Draw a circuit for p   q
r .
Solution
p   q r   p   q r 
~p
q
~r
© 2008 Pearson Addison-Wesley. All rights reserved
3-3-16
```