Exam 1 CMSC 203 Fall 2004
1. Circle T for True or F for False as they apply to the following statements:
T F In logic, all compound statements are either a tautology or a contradiction.
T F The Rational numbers are all infinite decimal expansions.
T F A set with 10 elements has 1024 subsets.
T F Sets that are disjoint have no elements in common.
T F H(010011000111, 110110110110) = d(100101110001).
T F The Rationals are closed under addition and multiplication.
T F The Irrationals are closed under addition and multiplication
T F If there is an ONTO function mapping set A to set B, then B is a subset of A.
T F Algorithms of polynomial order complexity are better than those of exponential
order complexity.
T F That all dogs like bones and Spot likes bones implies Spot is a dog.
2. Use the Laws of Logic to show: (p ∧ r) → q ≡ p → (r → q)
3. Find the inverse for the Universal Conditional Statement:
Every integer that is even has a square that is even.
4. Let f be function defined as: f = {(0, 3), (1, 2), (2, 4), (3, 1), (4, 0)}. Find f o f o f .
5. Let A = {1, 2, 4} and B = {0, 1, 2, 3, 4} and U = {0, 1, 2, 3, 4, 5, 6}
(a) Find A x (Bc ) (b) Find the Power Set of A (c) Find (A ∪ B)c ∩ (Ac − B)
6. Find the Big-O for the algorithm with complexity: n4(3n3 + 5) + n5(2n2 + 8n)
7. Show that the function f: R - R given by f(x) = 7x + 4 is a bijection.
8. Use Valid Reasoning to obtain the given conclusion: q ∨ ¬r ∨ t
s → p
r ∧ s
q → ¬p
-----------
∴ t
9. Prove TWO of the THREE theorems below:
Theorem 1: (by Contradiction) There is no greatest integer.
Theorem 2: The product of 3 consecutive integers is an even integer.
Theorem 3: (by Contaposition) For all integers, n, if n2 is odd, then n is odd.
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