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CS 70
Discrete Mathematics for CS
Spring 2008
David Wagner
Note 2
Lesson Plan
In order to be fluent in mathematical statements, you need to understand the basic framework of the language
of mathematics. This first week, we will start by learning about what logical forms mathematical theorems
may take, and how to manipulate those forms to make them easier to prove. In the next few lectures, we will
learn several different methods of proving things.
Propositions
A
proposition
is a statement which is either true or false.
These statements are all propositions:
(1)
√
3 is irrational.
(2) 1
+
1
=
5.
(3) Julius Caesar had 2 eggs for breakfast on his 10
th
birthday.
These statements are clearly not propositions:
(4) 2
+
2.
(5)
x
2
+
3
x
=
5.
These statements aren’t propositions either (although some books say they are). Propositions should not
include fuzzy terms.
(6) Arnold Schwarzenegger often eats broccoli. (What is “often?”)
(7) George W. Bush is popular. (What is “popular?”)
Propositions may be joined together to form more complex statements. Let
P
,
Q
, and
R
be variables rep
resenting propositions (for example,
P
could stand for “3 is odd”). The simplest way of joining these
propositions together is to use the connectives “and”, “or” and “not.”
(1)
Conjunction
:
P
∧
Q
(“
P
and
Q
”). True only when both
P
and
Q
are true.
(2)
Disjunction
:
P
∨
Q
(“
P
or
Q
”). True when at least one of
P
and
Q
is true.
(3)
Negation
:
¬
P
(“not
P
”). True when
P
is false.
CS 70, Spring 2008, Note 2
1
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View Full DocumentStatements like these, with variables, are called
propositional forms.
If we let
P
stand for the proposition
“3 is odd,”
Q
stand for “4 is odd,” and
R
for “5 is even,” then the propositional forms
P
∧
R
,
P
∨
R
and
¬
Q
are false, true, and true, respectively. Note that
P
∨¬
P
is always true, regardless of the truth value of
P
. A
propositional form that is always true is called a
tautology
, a statement that is always true regardless of the
truth values of its variables.
P
∧¬
P
is an example of a
contradiction
, a statement that is always false.
A useful tool for describing the possible values of a propositional form is a
truth table
. Truth tables are the
same as function tables. You list all possible input values for the variables, and then list the outputs given
those inputs. (The order does not matter.)
Here are truth tables for conjunction, disjunction and negation:
P
Q
P
∧
Q
T
T
T
T
F
F
F
T
F
F
F
F
P
Q
P
∨
Q
T
T
T
T
F
T
F
T
T
F
F
F
P
¬
P
T
F
F
T
The most important and subtle propositional form is an
implication
:
(4)
Implication
:
P
=
⇒
Q
(“
P
implies
Q
”). This is the same as “If
P
, then
Q
.”
Here,
P
is called the
hypothesis
of the implication, and
Q
is the
conclusion
.
1
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 Spring '08
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