CMSC203 - Discrete Structures - Final Examination - Spring 1999
Part One
1. Using the Properties of Sets, show that if A and B are sets, then A ∩ (A ∪ B) = A.
2. Find the Power Set of {1,{1}}.
3. Show that if p,q, and r are statements, then p → (q ∧ r) ≡ (p → q) ∧ (p → r).
4. Show the following is a valid argument:
p → q
r ∨ s
r → t
~q
s → p
∴ t
5. Find the Disjunctive Normal Form of the Boolean Polynomial F(x,y,z) = xy’ + z
6. What is the negation of the statement: All integers that are even are divisible by 2.
7. What is universal modus tollens? Draw a picture to illustrate it.
8. Prove one of the following statements:
a. The square root of an irrational number is irrational. b. If n is an odd integer, then 8 | (n2 − 1).
9. Use the Division Algorithm to find 37 MOD 5 and 37 DIV 5.
10. Use the Euclidean Algorithm to find GCD(274,136).
Part Two
11. Rewrite as a summation from 0 to 9.
12. Prove one of the two statements by Mathematical Induction:
a. If n is a positive integer, . b. Every positive integer has a binary representation.
13. Draw the directed graph of the relation R = {(a,b) | a,b ∈ {1,2,3,4,5,6} and a + b is even}
14. Show the relation Congruence Modulo 7 is an Equivalence Relation.
15. If f:{1,2,3,4} → {5,7,9,11} and g:{1,3,5,7,9,11} → {2,10,26,50,82,122} are given by f(x) = 2x + 3
and g(x) = x2 + 1, calculate (g ° f)−1
16. Let f:Z → Zeven be given by f(x) = 2x + 4. Show f is a bijection.
Part Three
17. How many 8-character license plates made up of 26 letters and 10 digits begin with “A1” and end
with “Z9”?
18. How many ways can 2 pennies, 3 nickels, 4 dimes, and 5 quarters be ordered in a line?
19. How many integer solutions are there to the equation: a + b + c + d + e + f = 50 with
a ≥ 2, b ≥ 4, c ≥ 6, d ≥ 1, e ≥ 3, and f ≥ 5?
20. TRUE or FALSE? C(7,0) + C(7,1) + ... + C(7,7) = 128.
21. Use the Iteration Method to solve sn = 7sn−1 + 1, when s0 = 1.
22. Find s152637 given sn = 3sn−1 + 28sn−2 and s0 = 2 and s1 = 5.
i10 i–
i1=
10
∑
7i
i0=
n
∑7n1+ 1–
6
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