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ICS 253 : Discrete Structures 1
Adel F. Ahmed
January 18, 2009(081)
St.Name:
St.Number:
Major 2
Time: 120 minutes
Question 1:
Answer the following: [2 points each]
•
If
f
:
R
→
R
and
f
(
x
) =
x
2
+ 1. Is
f
a bijection?
•
How many strings are there of lowercase letters of length
four or less?
•
Is
f
◦
g
=
g
◦
f
?
Question
Max Pts
Points
1
28
2
12
Total
40
•
If the successor of a set
A
is de±ned as
A
∪ {
A
}
. Then ±nd the successor of
{∅
,
{∅}}
.
•
What is the coe²cient of
x
8
y
9
in the expansion of (3
x
+ 2
y
)
17
?
•
How many positive integers not exceeding 100 are divisible either by 4 or by 6?
•
Translate the quanti±cation
∀
x
∈
Z
(
x
2
>
0) into English.
•
How many positive integers between 100 and 999 inclusive are divisible by 3 or 4?
•
Find the truth set of
P
(
x
) : “
x
2
> x
”.
•
Find
f
◦
g
where
f
(
x
) =
x
2
+ 1 and
g
(
x
) =
x
+ 2. (show steps).
•
How many bit strings of length 10 over the alphabet
{
a,b,c
}
have either exactly three
a
s or
exactly four
b
s?
•
Let
F
be the function such that
F
(
n
) is the sum of the ±rst
n
positive integers. Give a
recursive de±nition of
F
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