Sample Exam 1 - Fall 1999 - CMSC 203 / Discrete Structures
Symbols: N denotes the Natural Numbers, Z denotes the Integers, Q denotes the Rational Numbers, I denotes
the Irrational Numbers, and R denotes the Real Numbers, P (A) is the Power Set of a set A.
1. Circle T if the statement is true or F if the statement is false.
TFI ∩ Z = {0}
TF{1, ∅} ∈ P ({1, ∅}).
T F The inverse of the statement, If you mow my lawn, then I will pay you $25
is the statement, I do not pay you $25 implies you do not mow my lawn.
T F The following is a valid argument: ~t ∧ r
~t → (q ∨ p)
p → ~r
q → s
∴ s
TFIf A = {xy, yx}, then A × A = {xyxy, xyyx, yxxy, yxyx}.
TFIf Σ = {xy, yx} is an alphabet, then xyxyyxxyyx ∈ Σ5.
T F There is no prime number that divides both a positive integer and its successor.
T F If A and B are sets, then A ∪ (A ∩ B) = A.
TFIf a,b,q, and r are integers with 0 < r < b and a = bq + r, then gcd(a,b) = gcd(q,r).
T F The Disjunctive Normal Form of the Boolean Polynomial F(x, y) = 1 is
F(x, y) = xy + xy’ + x’y + x’y’.
2. Use the Euclidean Algorithm to find gcd(840,144).
3. a. Negate the statement: The square root of any positive integer is positive.
b. If you apply the Converse Error to the argument: Every integer is rational AND q is rational,
what conclusion follows?
4. Show, without using truth tables, that p → ~(q ∧ ~r) ≡ (p ∧ q) → r.
5. Find the Truth Table and Disjunctive Normal Form of the following circuit:
6. For the sets A = {a,b,c}, B = {a,d,e} and Y = {0,1}, verify that (A − B) × Y = (A × Y) − (B × Y)
7. Prove 2 of the 4 theorems below, using the indicated method:
Theorem 1: For all integers, n, if n3 is odd, then n is odd. (By Contraposition)
Theorem 2: The only positive integer that divides any integer and its successor is 1. (By Contradiction)
Theorem 3: If a and b are distinct integers then there is a rational number between them.
Theorem 4: The difference of the squares of successive integers is an odd integer.
x
y
z
F(x, y, z)
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