Fall 2007 - Discrete Structures - CMSC 203 - FINAL EXAM
1. Fill in the blank(s).
(a) (39 div 6)[GCD(1260,324)] = ______. (Hint: Do problem 2 first!)
39 mod 6
(b) ({1,3,4,7,9} − {2,3,4,5}) ∪ ({4,5,6,7} ∩ {1,2,3,4,5,6}) = ____________________
(c) H(1110001,1001001) = _________ (H is the Hamming Distance Function)
(d) If ∑ = {x,y,z}, then ⎜∑4⎜ = _________.
(e) DeMorgan’s Law states that for any sets A and B, (A ∩ B)c = _____________________________________.
(f) For finite sets A and B, the Inclusion/Exclusion Rule says:
⎜A ∪ B⎜ = _________ +_________ − ______________.
(g) If ∑ = {0,1}, then ∑ × ∑ = _______________________________________________________________.
(h) If ∑ = {0,1}, then ∑2 = __________________________________________________________________.
(i) Let F:{1,2,3,4} → {a,b,c,d} be a one-to-one correspondence. Then F = {(1,b),(2,d),(3,a),_______}
(j) If G:{x,y,z} → B is onto, and G = {(x,1),(y,2),(z,0)}, then B = _____________.
(k) If sn = 3sn−1 + 4sn−2 with s0 = 0 and s1 = 1, then s4 = _________.
(l) If f:R → R is defined by f(x) = (2x − 7)/3, then f −1(x) = ________________.
2. Use the Euclidean Algorithm to find GCD(1260, 324).
3. Find the truth table of the statement [p ∨ (q ∧ ~r)] → ~q.
4. Find the Disjunctive Normal Form of the Boolean Polynomial, f(x,y,z) = xy’.
5. Find the negation of: For all n ∈ Z, if n is prime, then n is not divisible by 2
6. Circle the validity of the following arguments:
(a) All dogs like swimming and Rover likes swimming, therefore Rover is a dog. VALID INVALID
(b) All dogs like swimming and Rover is a dog, therefore Rover likes swimming. VALID INVALID
(c) All dogs like swimming and Rover does not swim, therefore Rover is not a dog. VALID INVALID
7. Determine whether the relation R = {(x, y) | x, y are Integers and x divides y} is REFLEXIVE, SYMMETRIC,
TRANSITIVE, or none of these.
8. A youth group is made up of 22 boys and 30 girls. How many different ways can:
(a) they form a line? (b) they form a circle?
(c) they pick 5 boys and 7 girls to race each other if a certain pair of boys cannot be picked together?
(d) I have a large collection of $1 coins, and I want to distribute them into 10 piles so that each pile has at least 3
coins, and the total value of the piles is $50. How many ways can I do this?
(e) How many orderings are there of the letters of the word ORGANIZATION ?
9. In a class with 24 children, what is the probability that a child does not play soccer and is not on the honor roll
if 14 children play soccer, 12 children are on the honor roll and 6 children both play soccer and are on the honor
roll?
10. Find the best polynomial order for the algorithm with complexity (n3log n + 3n2 +1)(n + log n)
11. Prove: If a and b are distinct integers then there is a rational number between them.
12. Prove: The product of a rational number and an irrational number is irrational.
13. Prove using Mathematical Induction: 7i
i0=
n
∑7n1+ 1–
6
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