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SHEN’S CLASS NOTES191
1
Class Notes for Discrete Structures I
CS 191
Jan. 10, 2005
Textbook:
Discrete Mathematics
6/E
By Richard Johnsonbaugh
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Chapter One
Logic and Proofs
Logic
is the study of reasoning.
It provides reasoning rules by which a new result (statement)
can be obtained from known results and
axioms.
1.1
Propositions
Definition 1
A proposition
is a declarative sentence that is either true or false, but not
both.
Example 1
The following sentences are propositions
(a)
The only positive integers that divide 7 are 1 and 7.
(b)
1 + 1 = 3
(c)
Earth is the only planet that contains life.
Example 2
The following sentences are not propositions:
(a)
Please wear seat belt whenever you are driving.
(b)
Have a nice day!
SHEN’S CLASS NOTES191
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We often use letters
p
,
q
,
r
,
s
, …, etc. to represent propositions.
Example 3
p
: 1+1 = 3.
(We use p to represent the proposition 1+1 = 3.)
The value of a proposition is called a truth value, which can
be T (true) or F (false). Obviously, the truth value of
p
in the
above example
is F.
Definition 2
A compound proposition
is formed from existing propositions using logical operations.
Commonly used logical operations are
∧
(
and
operation),
∨
(
or
operation),
¬
(
not
operation),
→
(
implication
operation),
↔
(
bidirectional
operation).
We will explain them one by one.
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(1) Let
p
and
q
be two propositions.
p
∧
q
is a compound proposition read as
p
and
q
.
p
∧
q
is also called the
conjunction
of
p
and
q
.
∧
is a
binary operator
called
and
operator.
The truth value of
p
∧
q
depends on the truth values of
p
and
q
. We often use a
truth table
to show how to determine
the truth value of
p
∧
q
from the truth values of
p
and
q
.
The truth table for
p
∧
q
is shown below.
Table 1
The truth table for conjunction,
∧
(
p
and
q
)
p
q
p
∧
q
T
T
T
T
F
F
F
T
F
F
F
F
As we can see that
p
∧
q
is true if and only if both
p
and
q
are true.
SHEN’S CLASS NOTES191
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Example 4
p
:
It is raining
q
:
It is cold.
p
∧
q
:
It is raining and it is cold.
p
∧
q
is true if and only if both events are true: it is raining
and it is cold.
(2) Let
p
and
q
be two propositions.
p
∨
q
is a compound proposition read as
p
or
q
.
p
∨
q
is also called the
disjunction
of
p
and
q
.
∨
is a binary operator called
or
operator.
The truth table for
p
∨
q
is as follows.
Table 2
The truth table for disjunction,
∨
(
p
or
q
)
p
q
p
∨
q
T
T
T
T
F
T
F
T
T
F
F
F
Obviously,
p
∨
q
is true if and only if either
p
or
q
is true.
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Note
.
p
∨
q
will also be true if both
p
and
q
are true.
Therefore,
∨
is also called
inclusive or
.
Example 5
p
:
It is raining
q
:
It is cold.
p
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This note was uploaded on 04/12/2008 for the course CS 191 taught by Professor Shen during the Fall '06 term at University of MissouriKansas City .
 Fall '06
 Shen

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