CS 241, Sect
001
Homework 1
Dr. David Nassimi
Foundations I
Due: Week3, Fri. Sep. 18
Fall 2009
Chapter 1: Sets and Logic
1. Let
A
=
{
1
,
3
,
5
}
and
B
=
{
2
,
3
,
4
}
. Determine each of the following sets.
•
A
∪
B
•
A
∩
B
•
A

B
•
B

A
•
(
A
∪
B
)

(
A
∩
B
)
•
(
A

B
)
∪
(
B

A
)
2. Prove the following setequality, by using each of the methods described below.
(
A
∪
B
)
∩
(
A
∩
B
)
= (
A
∩
B
)
∪
(
A
∩
B
)
(a)
Proof using a Venn Diagram.
(b)
Algebraic proof.
This method uses setalgebraic manipulations to prove the equality, using
valid algebraic rules. The basic rules listed in the table below are assumed to have been proven,
and thus are valid rules to use.
Basic Rules (Equalities) in Set Algebra
Associative Laws:
(
X
∪
Y
)
∪
Z
=
X
∪
(
Y
∪
Z
)
(
X
∩
Y
)
∩
Z
=
X
∩
(
Y
∩
Z
)
Commutative Laws:
X
∪
Y
=
Y
∪
X
X
∩
Y
=
Y
∩
X
Distributive Laws:
X
∩
(
Y
∪
Z
) = (
X
∩
Y
)
∪
(
X
∩
Z
)
X
∪
(
Y
∩
Z
) = (
X
∪
Y
)
∩
(
X
∪
Z
)
De Morgan’s Laws:
X
∩
Y
=
X
∪
Y
X
∪
Y
=
X
∩
Y
Complement Laws:
X
∪
X
=
U
X
∩
X
=
φ
X
=
X
Repetition:
X
∪
X
=
X
X
∩
X
=
X
0/1 Laws:
φ
=
U
U
=
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 Spring '09
 NASSIMI
 Logic, Dr. David Nassimi, i. ii. iii.

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