This preview shows pages 1–2. Sign up to view the full content.
ArsDigita University
Month 2:
Discrete Mathematics  Professor Shai Simonson
Problem Set 2 – Sets, Functions, BigO, Rates of Growth
1. Prove by formal logic:
a. The complement of the union of two sets equals the intersection of the complements.
b. The complement of the intersection of two sets equals the union of the complements.
c. (
B

A
)
∪
(
C

A
) = (
B
∪
C
) –
A
.
d. If two sets are subsets of each other then they are equal.
2. Generalize De Morgan’s laws for
n
sets and prove the laws by induction.
3. Prove by induction on the size of the set, that the power set
P(A)
has cardinality
2
A

.
4.
A
⊕
B
is defined to be the set of all elements in
A
or
B
but not in both
A
and
B.
a. Determine whether or not
⊕
is commutative.
Prove your answer.
b. Determine whether
⊕
is associative.
Prove your answer.
c. Determine whether
⊕
can be distributed over union.
Prove your answer.
d. Determine whether
⊕
can be distributed over intersection.
Prove your answer.
5. Assume a universal set of 8 elements. Given a set
A = a
7
a
6
a
5
a
4
a
3
a
2
a
1
a
0
,
represented by 8
bits, explain how to use bitwise and/or/not operations, in order to:
a. Extract the rightmost bit of set
A
.
b.
Make the odd numbered bits equal 0.
c. Make bits 46 equal to 1.
Given another set
B,
explain how to: a.
Determine if A
⊆
B.
b.
Extract A

B.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '09

Click to edit the document details