# CSE 504 Discrete Structures

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```					CSE 504 Discrete Structures & Foundations of Computer Science
Instructor: Dr. Djamel Bouchaffra
Date: February 19, 2003

Regular Exam

Exercise #1 (Section 1.3 #23 p.41)
Translate each of these statements into logical expressions using predicates, quantifiers and logical
connectives.
a. No one is perfect
b. Not everyone is perfect
c. All your friends are perfect
d. One of your friends is perfect
e. Everyone is your friend and is perfect
f. Not everybody is your friend or someone is not perfect.

Exercise #2 (Section 1.5 # 21 p.75)
Prove that if n is an integer and n3 + 5 is odd, then n is even using
a. An indirect proof

Exercise #3 (Section 1.8 #37 p.110)
1
Let f be a function from A to B. Let S be a subset of B. Show that f        ( S )  f  1( S ) .

Exercise #4 (Section 204 #53 p,168)
Encrypt the message “DO NOT PASS GO” by translating the letters into numbers, applying the
encryption function given, and then translating the numbers back into letters.

a.   f(p) = (p + 3) mod 26 (the Caesar cipher)
b.   f(p) = (p + 13) mod 26
c.   f(p) = (3p + 7) mod 26

Exercise #5 (Section 2.6 #9 p.180)
Convert (1011 0111 1011)2 from its binary expansion to its hexadecimal expansion.

Exercise #6 (Section 2.2 #63 a, b, c, d, & e p.144
(Calculus required) For each of these pairs of functions, verify whether f = O(g)

a.   f(x) = x2 +3x +7
g(x) = x2 + 10
b.   f(x) = x2log x
g(x) = x3
c.   f(x) = x4 + log(3x8 +7)
g(x) = (x2 + 17x +3)2
d.   f(x) = (x3 + x2 + x +1)4
g(x) = (x4 + x3 + x2 + x +1)3
e.   f(x) = log (x2 + 1)
g(x) = log x

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