CMSC 203 Fall 2003 Final Examination Page 1 Name____________________________
1. Use the Laws of Logic to show: p → (¬q → ¬r) ≡ r → (p → q)
2. Negate: For all Integers, n, if n is positive, then n2 is positive.
3. Find the Contrapositive form of: Some people who are fast runners win races.
4. Find the Big-O of the algorithm with complexity:
(3x5 + 2x2 + 1)(4x2 + 1) + (5x4 + 3x2 + 7)(2x3).
5. What is the probability that a binary string of length 8 will have at least six 1’s?
6. How many ways can I fill a cooler with cans of soda if the cooler holds 30 cans,
I have 8 different types of soda, and I want at least 2 of each type in the cooler?
7. Graph the relation R = {(a,b) | a,b ∈ {0, 1, 2, 3, 4, 5, 6, 7} and b = (a2 mod 5)}.
8. Find the matrix of the relation R in Question 7.
9. Let f = {(1,3), (2,2), (3,4), (4,5), (5,1)}, g = {(1,5), (2,1), (3,4), (4,2), (5,3)},
and h = {(1,4), (2,5), (3,1), (4,2), (5,3)}. Find h o g o f.
10. Find the Disjunctive Normal Form of the Boolean polynomial F(w,x,y,z) = w’xy’ + z
11. Find the next 5 terms of sn = 2sn−1 − 3sn−2 when s0 = (−1) and s1 = 1.
12. How many ways can I line up 3 pennies, 5 nickels, 2 dimes, 8 quarters, and 6 half-dollars,
if all the coins are from 1990?
13. Graph an example of a Reflexive and Anti-symmetric relation on the set {1, 2, 3, 4}.
14. Graph an example of a One-To-One function that is NOT Onto.
15. Prove one of the following theorems by Induction:
Theorem: For all Integers n > 0 and a ≠ 0,1, .
Theorem: A set with n elements has 2n subsets.
16. Prove one of the following theorems by Contradiction:
Theorem: is irrational.
Theorem: If every Integer greater than 1 is divisible by a prime, then the set of prime numbers is
infinite.
17. Prove one of the following theorems:
Theorem: If f(x) = x2 + 1, then S = {(x,y) | x,y ∈ R and f(x) = f(y)} is an Equivalence Relation.
Theorem: If g(x) = 3x −2, then g is a Bijection from R to R.
ai
i0=
n
∑an1–
a1–
--------------=
2
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