CMSC 203 - Discrete Structures - FINAL EXAMIMATION - Fall 2005
1. (a) Use the Laws of Logic to show: ¬(p ↔ q) ≡ p ⊕ q
(b) Find the negation of the following Universal Conditional
Anyone who sings is in the choir.
(c) Use the rules of inference to show the following is a valid argument:
p → q
r → t
¬q
r ∨ s
s → p
∴ t
2. (a) Find A × B for the sets A = {100, 010, 001} and B = {10, 11, 01}
(b) Use the Properties of Sets, to show (A ∪ B) − (A ∩ B) = (A − B) ∩ (B − A).
3. (a) What Domain and Image make the following F a One-to-One Function:
F = {(3, 2), (1, 5), (5, 4), (6, 3), (7, 9), (8, 8)}?
(b) What is the inverse of F above?
(c) Find F(G(x)) for the following Real-valued functions: F(x) = 4(3x-5) and G(x) = 5x2 + 7.
4. (a) Use the Euclidean Algorithm to find GCD(144,68).
(b) Find Big-O of the algorithm whose complexity is F(n) = (n6)(n7 + n5)(n4 + n2 + 1).
5. Prove 1 of the 2 Theorems below:
Theorem 1: If a prime integer divides the square of an integer, then it divides the integer.
Theorem 2: The square root of a prime integer is irrational. (Assume Theorem 1)
6. (a) Find the next 3 terms of the sequence sn = sn-1 − sn-2 when s0 = 1 and s1 = 1.
(b) Find an expression for the series:
7. Prove 1 of the 2 Theorems that follow by Mathematical Induction.
Theorem 1: There are 2n binary strings of length n.
Theorem 2: If sn = sn-1 + sn-2 + sn-3 when s0 = 1, s1 = 3, and s2 = 5, then sn is odd, for all n > 3.
53()
i7–
i0=
1000
∑
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