Fall 2000 CMSC203 Exam 1
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.
TFQ − Z = I.
TF{∅} ∈ P ({∅, U}). .
T F The negation of the statement, No prime numbers are even is the statement, All prime numbers
are even.
T F The following is a valid argument: s → ~q
~p → (r ∨ s)
~p ∧ q
r → t
∴ t
T F If A = {0,1} and B = {2,3}, then A × B = {(0,0),(0,1),(1,0),(1,1),(2,2),(2,3),(3,2),(3,3)}.
TFIf Σ is an alphabet and m < n are positive integers, then Σm ⊆ Σn.
T F According to Universal Modus Ponens, the argument: All that glitters is gold AND my watch
glitters THEREFORE my watch is gold is valid.
T F If A and B are sets, then A ∩ (B ∪ A) = B .
TFIf a,b and c are integers with a = b + c, then GCD(a,b) = GCD(b,c).
T F For any Boolean Polynomial, its Disjunctive Normal Form and Conjunctive Normal Form are
negations of each other.
2. Using the Laws of Logic, show that (~p ∧ q) → (p ∨ ~r) ≡ (q ∧ r) → p
3. Given the informal language statement: Every integer that is prime is odd
a. Rewrite the statement in formal language as a Universal Conditional.
b. In Formal Language, find the Inverse Form of the statement.
c. In Formal Language, find the negation of the statement.
4. Calculate: a. 217 div 6 b. 217 mod 6 c. GCD(217,6) using the Euclidean Algorithm
5. Find the Disjunctive Normal Form of the Boolean Polynomial F(x,y,z) = x + x’z.
6. For the sets A = {a,b,c,d}, B = {a,c,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 the Method of Contraposition)
Theorem 2: If a, b and c are positive integers such that a = b + c and GCD(a,b) = GCD(b,c),
then GCD(a,c) = GCD(a,b) = GCD(b,c).
Theorem 3: If A, B, and C are sets, then (A ∪ B) − C = (A − C) ∪ (B − C). (By the Properties of Sets)
Theorem 4: The difference of the squares of successive integers is an odd integer.
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