Fall 2010 Examination 1 CMSC 203
1. Circle T for True or F for False as they apply to the following statements:
T F If a statement is a tautology then its negation is a contradiction.
T F A set with 6 elements has 36 non-empty subsets.
T F The empty set is a subset of all sets.
T F The Range and Image of an Onto function are the same set.
T F If Σ
= {0, 1} , then |Σ
6
| = 2
6
.
T F The negation of an implication is a disjunction.
T F The densities of a binary string and its negation are equal.
T F H(00001111, 10101010) = 4.
T F The converse and inverse of a conditional statement are logically equivalent.
T F The set of all finite-length binary strings is countable.
2. Use the Laws of Logic to show: p
∨
(
¬
q
→
r) ≡ r
∨
(
¬
p
→
q)
3. Find the negation of the following Universal Conditional: Some people who run fast win races.
4. Use the Rules of Inference to show the following is a valid argument:
p
∧
¬
r q
→
r (
¬
q
∧
s)
→
¬
p
∴
¬
s
5. Sort the set of binary strings of length 4 by their densities, smallest to largest.
6. Using the Properties of Sets, to show A − (B
∪
C) = (A − B)
∩
(A − C).
7. Given the function F = {(1, 2), (2, 1), (3, 2), (4, 1), (5, 2)}
(a) What is the Domain of F? (b) What is the Image of F?
(c) What is the Inverse of F? (d) Why or why not is the Inverse in (c) a function?
8. Find F
°
F
°
F
°
F for F: {0, 1, 2, 3, 4}
→
R given by F(x) = 2x
−
1.
9. (8 points) For the given argument, circle MP if it is an example of Modus Ponens, MT if it is an
example of Modus Tollens, CE if it is an example of Converse Error, and IE if it is an example of
Inverse Error.
MP MT CE IE All boys like football and Paul likes football, therefore Paul is a boy.
MP MT CE IE All boys like football and Paul is a boy, therefore Paul likes football.
MP MT CE IE All boys like football and Paul is not a boy, therefore Paul dislikes football.
MP MT CE IE All boys like football and Paul dislikes football, therefore Paul is not a boy.
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