Fall 2007 - CMSC203 - Discrete Structures - Exam 1
1. Circle T for True or F for False as they apply to the following statements:
T F Tautology and contradiction are logical negations of one another.
T F A Conditional Statement and its Converse are logically equivalent.
T F The Empty (Null) set is a subset of itself.
T F Half the subsets of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} contain the element 3.
T F For any sets A and B, (A ∪ B) ⊆ (A x B).
T F The set {1, 2, 3, 4, 5, 6} has 64 non-empty subsets.
T F For any non-empty sets A and B, |A x B| = |B x A|.
T F If F:A → B is a function, then F(A) ⊆ A.
T F The Cardinality of the Rationals and the Irrationals is the same.
T F If s is a binary string of length 8 and t = (s ⊕ 00000000), then s = t.
2. Find the truth table for the compound statement: [( p ↔ q) ∨ r] ⊕ [p ∧ (¬q → r)]
3. Find the related forms for the statement: For all Integers, x, if x is odd, then (x + 2) is odd.
CONVERSE: __________________________ NEGATION: ___________________________.
4. Draw a graph for a function, f:{1, 2, 3, 4}→ {w, x, y, z}, that is: (a) onto; (b) one-to-one.
5. Show that the function f : R → R defined as f(x) = 5x − 12 is a bijection.
6. Calculate the following (assuming all strings are from the alphabet {0, 1}):
(a) |Σ8|(b) d(010010010010)(c) H(1111011101 , 0011010011)(d)
7. (a) Let f = {(0, c), (1, a), (2, e), (3, d), (4 ,b)} and g = {(a, 2), (b, 3), (c, 0), (d, 1), (e, 4)}.
Show that ( g ° f ) −1 = f −1 ° g −1.
(b) Find the Inverse of the function of h = {(1, 5), (2, 4), (3 ,1), (4, 4), (5, 1)}.
8. Use the logic of valid arguments to determine whether or not we can deduce t:
p → ¬q
r ∨ q
¬r ∧ s
¬p → t
9. Use the Properties of Sets to verify for any sets A and B: (Ac − B)c = A ∪ B
0.33 1–
(
)1.33–1+()
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