Exam 1 - Discrete Structures - Spring 2004
1. Circle T for True or F for False as they apply to the following statements:
T F In logic, disjunctions and implications are equivalent.
T F The Rational numbers have the same cardinality as the Real numbers.
T F Every set has the Empty Set as a subset.
T F If F:A → B, is a function, then F(A) and B have the same cardinality.
T F {1, 2, 3, 4, 5} and {6, 7, 8, 9, 10} are disjoint.
T F If n is an integer, then floor(n) = ceiling(n).
T F DeMorgan’s Law for Logic states that the negation of a disjunction
is a conjunction.
T F If there is an bijective function mapping set A to set B, then A = B.
T F A statement and its converse are NEVER logically equivalent.
T F Each child likes candy and Sandy likes candy implies Sandy is a child.
2. Use the Laws of Logic to show: p → (q ∨ r) ≡ (p → q) ∨ (p → r)
3. Find the contrapositive 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, u), (3, i), (4, o)}, and
g = {(a, 4), (e, 1), (i, 0), (o, 3), (u, 2)}.
Show, computationally that ( f o g )−1= g−1 o f −1
5. Let A = {d, i, s, c} and B = {r, e, t} and U = {a, c, d, e, h, i, m, r, s, t}
(a) Find A x B (b) Find the Power Set of B (c) Verify the (A ∪ B)c = Ac ∩ Bc
6. Circle the best choice to complete the statement:
The argument “All men are mortal and Socrates is not a man, therefore Socrates is not mortal” is
an example of CONVERSE INVERSE ABDUCTION error.
7. Given the sets A = {1, 2, 3, 4}, B = {5, 6, 7, 8, 9}, and C = {10, 11, 12, 13}, draw the directed
graph of a function with the following properties:
(a) mapping A to B that is NOT one-to-one.
(b) mapping B to C that is onto.
(c) mapping A to C that is neither one-to-one nor onto.
8. Prove TWO of the THREE theorems below:
Theorem 1: The function F: R → R, defined as F(x) = 8x − 15, is a bijection.
Theorem 2: The sum of 4 consecutive integers is an even integer.
Theorem 3: (by Contradiction) The sum of Rational and Irrational numbers is Irrational.
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