Fall 2003 Examination 1 CMSC 203
1. Circle T for True or F for False as they apply to the following statements:
T F Every conditional statement is logically equivalent to a disjunctive statement.
T F Every rational number is equivalent to an infinite, repeating decimal.
T F No set has itself as a subset.
T F The domain and image of a one-to-one function have the same cardinality.
T F Disjoint sets have empty intersections.
T F The Floor of the sum of two integers is equal to the sum of the Floors of the numbers.
T F Two sets that are subsets of each other are equal.
T F If there is an injective function mapping set A to set B, then A = B.
T F A conditional statement and its contrapositive are logically equivalent.
T F The negation of an Existential statement is a Universal statement.
2. Find the truth table for the compound statement: p → ¬[q → (r ∨ p)]
3. Find the negation for the Universal Conditional Statement:
Every integer that is greater than 1 has a unique prime factorization.
4. Let f and g be functions defined as:
f = {(0,e), (1,a), (2,c), (3,h), (4,n), (5,i), (6,g), (7,t)}, and
g = {(a,2), (c,4), (e,6), (g,0), (h,1), (i,3), (n,5), (t,7)}.
Show, computationally that ( f o g )−1= g−1 o f −1
5. Let A = {1, 2, 5, 7} and B = {2, 3, 6} come from the Universal Set {1, 2, 3, 4, 5, 6, 7, 8}
(a) Find A x B (b) Find the Power Set of B (c) Verify the (A ∪ B)c = Ac ∩ Bc
6. Calculate the following:
(a) L(101110000) (b) d(00110011)
(c) H(000111000111, 111111000000) (d)
7. Verify the following valid argument: p ∧ s
t → ¬p
t ∨ r
(s ∨ r) → q
∴ q
8. Prove that the function F: R → R, defined as F(x) = 7x + 11, is a bijection.
Show F is 1-1:
Show F is Onto:
Conclusion:
5.3()–1+()5.3()
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