CMSC203 - Spring 2006 - Examination 1
1. Circle T for True or F for False as they apply to the following statements:
T F In Logic, Conditional statements are logically equivalent to a Disjunctive statement.
T F The argument “All dogs bark and Ralph is not a dog, therefore Ralph does not bark” is
an example of Modus Tollens.
T F Only the Empty set is a subset of itself.
T F If the set A has |A| = n and a is in A, then A has 2n − 1 subsets containing a.
T F For any sets A and B, (A ∩ B) ⊆ A ⊆ (A ∪ B).
T F The statements p → q and q → p are inverses of one another.
T F If the set |A| = n then |A x A x A| = 3n .
T F Injective functions have inverses that are Injective functions.
T F The Cardinality of the Integers and the Reals is the same.
T F For an alphabet Σ, the set Σ* contains no strings of infinite length.
2. Find the truth table for the compound statement: [( p ∨ ¬q) ↔ r] ⊕ [p → (q ∧ r)]
3. Find the related forms for the statement: Some people who ice skate play hockey.
INVERSE: _________________________________________________________________.
NEGATION: __________________________________________________________________.
4. Draw a graph for a function, f:{1, 2, 3, 4}→ X, that is: (a) onto; (b) one-to-one. (You pick X)
5. Show that the function f : R → R defined as f(x) = 7 − 3x is a bijection.
6. Calculate the following (assuming all strings are from the alphabet {0, 1}):
(a) l(101010010101) (b) d(00000000)
(c) H(10010011 , 01010101) (d)
7. (a) Let f = {(m,g),(n,f),(o,h),(p,d),(q,e)} and g = {(d,q),(e,o),(f,m),(g,p),(h,n)}.
Show that ( g ° f ) −1 = f −1 ° g −1.
(b) Find the Inverse of the function of g.
8. Use the logic of valid arguments to determine whether or not Bob is happy:
1. If Ann is happy, then Bob is sad..
2. If Carl is happy or Deb is happy, then Ed is sad.
3. Ed is happy and Carl is sad.
4. If Deb is sad, then Ann is happy.
9. Use the Properties of Sets to verify for any sets A, B, C, and D
(A ∩ B) − (C ∩ D) = [(A − C) ∩ (Β − C)] ∪ [(A − D) ∩ (B − D)]
5.3()–5+()5.3 5–()
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