CIS 66 – Midterm 1 Last Name (print)
Spring 2002
ssn (last 4 digits)
First (Name print)
____________________________
__ __ __ __
____________________________
Signature
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1. (20 points) Compute the following - the result must be a number
log2(64*1024)
_________
log4(64)
_________
The bitwise result of ((1000)  (1010))  (0101)
_________
200

j = 21
J
_________
If set A = {{a, b, c}}, what is |A|
_________
If set A = {{a, b, c}}, what is |P(A)|
_________
If set A = {{a, b, c}} and set B = {1, 2, 3}, how many
elements are in the set AB?
_________
Compute -94/10
_________
The prime factorization of 36.
__________________________
The prime factorization of 3600.
__________________________
GCD(1000001, 1000) (that’s 1,000,001 and 1,000)
_________
GCD(22*3*5*7, 2*32*5)
_________
LCM(22*3*5, 2*32*5)
_________
Convert A11 (base 16) to decimal.
_________
Convert A11 (base 16) to binary.
____ ____ ____
1. (20 points) Compute the following - continued
Convert 65 decimal to binary.
_________
Convert 65 decimal to base 8.
_________
-45 mod 100
_________
What decimal number does the 8-bit,
2’s complement number 10100001 represent?
_________
What decimal number does the 8-bit,
1’s complement number 10100001 represent?
_________
2. (20 points) Circle the correct answer
propositions: If 1 + 1 = 3, then 1 + 1 = 2
True False
propositions:  x  y  Px,y   ( y  x Px,y)
True False
propositions:  jZ  kZ (j = k2)
True False
propositions: pp
tautology contradiction contingency
propositions: (pq)  (q  p) tautology contradiction contingency
propositions: (pq)  pq
tautology contradiction contingency
sets: {x}  {{x}}
True False
sets: {{}}  {{x}}
True False
sets: (1, a)  A  B where A = {1, 2}, B = {a, b}
True False
sets: A(BC)  (AB)AB
True False
Summations
N+1

j=3
j-1
=
n-1

k=1
k+1
True False
2. (20 points) Circle True or False, continued
-x = - x
True False
3x3 + 5x + 2 is O(x!)
True False
If a, b, and c are integers and a | b and a | c, then a| (b + c)
True False
If A and B are n by n matrices, AB = BA.
True False
-36 | -72
True False
1,000,001 and 100 are relatively prime.
True False
Problems that are polynomial worst-case complexity are
tractable.
True False
y = x + 1, xZ, yZ (the integers)
injection surjection
bijection
y = x*2, xZ, yZ (the integers)
injection surjection
bijection
3. (10 points) Create a truth table for the proposition
[(p  q)  (qr)]  (p  r)
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
4. (5 points) Define the following sets by listing their elements if A, B, C, and D
are defined as follows.
A = {a, b, c, d}
B = {a, c, e, g}
C = {b, d, f}
D = {a, b}
AC = {_______________________________________________}
AB = {_______________________________________________}
P(D) = {_______________________________________________}
CD = {_______________________________________________ }
AD = {_______________________________________________ }
5. (10 points) Find the indicated functions when f(x) and g(x) (from R to R) are
defined as follows:
f(x) = x2 + 2
g(x) = 2x + 3
(fg)(x)
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(ff)(x)
____________________________________
(fg)(x)
____________________________________
(f+g)(x)
____________________________________
(g-1)(x)
____________________________________
f(x) is an injection (one to one).
True
false
f(x) is an surjection (onto).
True
false
g(x) is an injection (one to one).
True
false
g(x) is an surjection (onto).
True
false
6. (10 points) Assume a, b, and c are integers. Show (prove) that, if a|b and a|c,
then a|(b+c).
7. (10 points) Assume matrices A, B, and C are defined as follows. Compute
AB – 2C.
A=
3
1
-2
2
B=
1
0
3
1
C=
2
2
-3
1
8. (10 points) Provide the best possible big O estimate (that is, a big Theta
estimate) for each of the following functions
f(x) = (3 x3 + 4x2+ 5 x + 2)*(x + log x)
______
f(x) = (3 x3 + 4x2+ 5 x + 2)/( log x + x2)
______
f(x) = (3 x3 + 4x2+ 5 x + 2)*( log xx + x2)
______
f(x) = (x!)*(xx)*(x)
______
f(x) = (xx + x2x + 5x)(x! + 5x)
______
9. (10 points) Provide the best possible big-O estimate for
f(x) = x3 + 5x2 log x + x2 + log xx. Then show (prove) that your answer is in fact a
big-O estimate.