Fall 2008 CMSC 203 Discrete Structures Exam 1
1. Circle T for True or F for False as they apply to the following statements:
T F The negation of a Conjunctive statement is a Disjunctive statement .
T F The negation of an Existential statement is a Universal statement.
T F Non-empty sets have at least 2 subsets.
T F If F:A → B is function, then F = A × B.
T F For any sets A and B, (A ∪ B) ⊆ (A ∩ B).
T F The empty set has no subsets.
T F N ⊆ Z ⊆ Q ⊆ R ⊆ C.
T F Finite sets are Countable.
T F That All cars have steering wheels and A motorcycle does not have a steering
wheel implies A motorcycle is not a car is an example of Universal Modus Tollens.
T F If s is a string over an alphabet Σ, and s is in Σn, then the density of s is n.
2. Find the truth table for the compound statement: [( p ⊕ r) ∨ q] → [p ∧ (¬r → q)]
3. Find the related forms of: For all Integers, x, if x is even, then x is a multiple of 2.
CONVERSE: ___________________. NEGATION:_____________________________.
4. Circle the appropriate response when the given argument illustrates Universal Modus Ponens
(MP), Universal Modus Tollens (MT), Converse Error (CE), or Inverse Error (IE), :
MP MT CE IE All cats eat mice and Tiger is a cat therefore Tiger eats mice.
MP MT CE IE All cats eat mice and Tiger is not a cat therefore Tiger does not eat mice.
MP MT CE IE All cats eat mice and Tiger does not eat mice therefore Tiger is not a cat.
MP MT CE IE All cats eat mice and Tiger eats mice therefore Tiger is a cat.
5. Show that the function f : Z → Z defined as f(x) = (x − 3) is a bijection.
6. Calculate the following (assuming all strings are from the alphabet {0, 1}):
(a) |{1, 2, 3} × {4, 5, 6, 7}|(b) d(10010101011001)
(c) H(001001100011 , 100101010010) (d)
7. (a) Let f = {(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)}. Find f ° f ° f ° f .
f ° f ° f ° f = {(0,___), (1,___), (2,___), (3,___), (4,___)}
(b) Find the Inverse of the function of f .
f −1 = {(0,___), (1,___), (2,___), (3,___), (4,___)}
8. Use the logic of valid arguments to determine whether or not we can deduce p:
p ∨ q ¬t r → s ¬s ∨ t q → r
9. Given a set A and a collection of non-empty subsets of A, {A1, A2, A3, ..., An}, we call this collec-
tion a partition of A provided:
(i) (A1 ∪ A2 ∪ A3 ∪ ... ∪ An) = A, and (ii) Ai ∩ Aj = ∅ when i ≠ j.
(a) What set is partitioned by the collection { {1, 3, 4}, {2, 5}, {7, 8, 9} }?
(b) Why is the collection of intervals of the Reals, {... , [−3,−2], [−2,−1], [−1,0], [0,1], [1,2], [2,3], ...}
NOT a partition of the Reals? (note: [a,b] = {x ∈ R | a ≤ x ≤ b}.
1.5–4+()3.7 4+()
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