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1
•
DeMorgan’s
Laws
¬
(
A
∪
B
)
=
¬
A
∩¬
B
¬
(
A
∩
B
)
=
¬
A
∪¬
B
A
∪
B
¬
(
A
∪
B
)
B
¬
B
∩
A
¬
A
=
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Example of using DeMorgan’s Law
Let
A
= {
a, d
,
e
,
g
}
and
B =
{
c
,
d
,
f
,
g
} from the universe
U
= {
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
,
i
}. To verify that
¬
(
A
∪
B
)
=
¬
A
∩¬
B
find each of these two sets independently to find that they
are indeed the same.
1)
¬
(
A
∪
B
)=
¬
{
a, c
,
d
,
e
,
f
,
g
} = {
b
,
h
,
i
}
2)
¬
A
∩¬
B=
{
b
,
c
,
f
,
h
,
i
}
∩
{
a
,
b
,
e
,
h
,
i
} ={
b
,
h
,
i
}
are the same, as predicted.
3
•
Domination Laws
A
∪
U
=
U
A
∩
∅
=
∅
•
Absorption Laws
A
∪
(
A
∩
B
) =
A
A
∩
(
A
∪
B
) =
A
p
∨
T
⇔
T
p
∧
F
⇔
F
p
∨
(
p
∧
q
)
⇔
p
p
∧
(
p
∨
q
)
⇔
p
These laws can be proved by using logic laws or
membership tables.
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In everyday life different kinds of proofs are acceptable:
•
Jury trial.
•
Word of God.
•
Word of Boss.
•
Experimental science: The truth is guesses and confirmed
or refuted by experiments.
•
Sampling: like public opinion is obtained by polling.
•
Inner conviction.
.
These are not valid proofs in mathematical sense.
They all can go wrong…
What is a proof?
A proof is a method of ascertaining truth.
5
Mathematics uses a particularly convincing way to argue
that something is true.
Definition
. A proof is a formal verification of a
proposition
by a chain of
logical deductions
starting from the base
set of
axioms
.
A proof takes axioms and definitions and uses deduction rules,
step by step, to get a desired conclusion.
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Proof methods
•
If a statement considers a few numbers of cases it can be proved
by
exhaustive checking
.
Example
: All students in this class are computer science major.
We can easily verify is it true or false.
•
Truth table method
. To prove a statement about small number
Boolean variables make a truth table and check all possible cases.
Example
: (
p
→
q
)
↔
(
¬
q
→
¬
p
).
By inspection we see that lhs has the same truth value as the rhs
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This document was uploaded on 07/14/2011.
 Spring '09

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