Fall 1997 Examination 1 CMSC 203
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.
TFR − Z = I.
TF∅ ∈ P({∅}). .
T F The inverse of the statement, I finished my work implies I went to a movie
is the statement, I did not go to a movie implies I did not finish my work.
T F The following is a valid argument: ~p ∧ q
~p → (r ∨ s)
s → ~q
r → t
∴ t
T F If A = {0,1}, then A × A = {00, 01, 10, 11}.
TFIf Σ = {goo, ga} is an alphabet, then googoogaga ∈ Σ10.
T F The set of prime numbers and the set of composite numbers partition the set of positive integers.
T F If A, B, and C are sets, then (A ∩ B ∩ C)c = Ac ∪ Bc ∪ Cc .
T F If a,b,q, and r are integers with 0 < r < b and a = bq + r, then gcd(a,b) = gcd(b,r).
T F The Disjunctive Normal Form of the Boolean Polynomial F(x, y, z) = x is F(x, y, z) = xyz + x’y’z’.
2. Use the Euclidean Algorithm to find gcd(330, 210).
3. Given the informal language statement: Every prime number is odd
a. Rewrite the statement in formal language as a Universal Conditional.
b. Find the negation of the statement (in either formal or informal language).
4. Show, without using truth tables, that p → (q ∧ r) ≡ (p → q) ∧ (p → r).
5. Find the Boolean polynomial representing a circuit of four switches controlling a light bulb in such a way that
if the middle two switches are the opposite of the first and last, then the bulb turns on.
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, then gcd(a,b) = gcd(b,c).
Theorem 3: If A, B, and C are sets, then (A ∪ B) − C = (A − C) ∪ (B − C).
Theorem 4: The difference of the squares of successive integers is an odd integer.
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