CMSC 203 Fall 2004 Final Examination
1. Use the Laws of Logic to show: p → ¬(q ∧ ¬r) ≡ (p ∧ q) → r
2. Negate: Some integers are perfect squares and odd..
3. Find the Contrapositive form of: All dogs that run fast have long legs.
4. Find the Big-O of the algorithm with complexity:
(3x6 + 4x3 + 1)(x4 + 2x) + (3x7 + 7x3 + 2)(2x3).
5. What is the probability that a binary string of length 10 will have no more than three 1’s?
6. How many ways can I fill a cooler with cans of soda if the cooler holds 60 cans,
I have 8 different types of soda, and I want at least 4 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 − 2) mod 5]}.
8. Find the matrix (MR o MR) of the relation on {1, 2, 3}:
R = {(1,1), (1,3), (2,1), (2,3), (3,1), (3,2), (3,3)}
9. Let f = {(1,3), (2,2), (3,4), (4,5), (5,1)}, g = {(1,4), (2,5), (3,1), (4,2), (5,3)},
and h = {(1,5), (2,4), (3,3), (4,2), (5,1)}. Find h o g o f.
10. Find the Disjunctive Normal Form of the Boolean Expression that describes a 3-way light
switch controlling a lightbulb that INCLUDES the case of being ON when the first two switches
are ON and the third is OFF.
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 5 Pennies, 2 Nickels, 4 Dimes, 3 Quarters, and 6 Half-dollars,
if all the coins are from the same year? (e.g. HQHDNDHQHDNQDHH)
13. Let E and F represent event sets within the sample space S. If there is a 50% probability of E
occurring within F and there is a 50% probability of F occurring within S, show the E is indepen-
dent of F.
14. Graph an examples of:
(a) an Onto function that is NOT One-To-One. (b) a One-To-One function that is NOT Onto.
Docsity.com
