CMSC 203 - Discrete Structures - FINAL EXAMIMATION - Spring 2006
1. (a) Use the Laws of Logic to show the BICONDITIONAL and EXCLUSIVE-OR are
negations of one another.
(b) Find the converse of the following Universal Conditional:
All people who like Mathematics get good jobs.
(c) Use the rules of inference to show the following is a valid argument:
I am rich implies I am happy.
My job is fun or I am not happy.
I am rich.
I do not ride a bicycle implies my job is not fun.
∴ I ride a bicycle
2. (a) Find A × B for the sets A = {0,1} and B = {11, 111, 1111}
(b) Using the Properties of Sets, show for any sets A, B and C, (A ∩ B) − C = (A − C) ∩ (B − C).
3. (a) For F = {(1,3), (4,2), (3,7), (7,4), (5,5)}, what Domain and Image make F a bijection?
(b) What is the inverse function of F in part (a)?
(c) Find F(G(x)) for the following Real-valued functions: F(x) = 2(x+2) and G(x) = x − 2.
4. (a) Use the Euclidean Algorithm to find GCD(155, 85).
(b) Find Big-O of the algorithm whose complexity is F(n) = (3n5)(n6 + 5n7)(2n3 + 3n + 2).
5. Prove 1 of the 2 Theorems below:
Theorem 1: A set with n elements has 2n subsets. (Hint: Compare to binary strings)
Theorem 2: The square root of an irrational is irrational.
6. (a) Find the next 3 terms of the sequence sn = (2sn-1)(sn-2 )− (n2) when s0 = 1 and s1 = 0.
(b) Evaluate the series: .
7. Prove 1 of the 2 Theorems that follow by Mathematical Induction.
Theorem 1: If Σ = {0,1}, then |Σn| = 2n.
Theorem 2: Is sn = sn-1 + sn-2 + sn-3 + sn-4 when s0 = s1 = s2 = s3 = 4 then 4 divides sn, for all n > 4.
8. (a) If a state issues license plates using the 26 letters {A, B, C, ..., Z}, how many distinct plates of
8 letters can they create that begin with C or end with G?
(b) How many plates can they create if they each plate can only have a letter appear no more than
once and no plate can contain the string CRASH?
4i2i–
i0=
1000
∑
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