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 of A equal 0.
c.
Make bits 46 of A equal to 1.
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 Spring '09
 Set Theory, Financial aid, Natural number, Countable set

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