CMSC 203 - Exam 1 - Spring 2002
Notation: Let R denote the Real Numbers, and P(A) denote the Power Set of A.
1. (20 pts.) Circle T if the statement is true or F if the statement is false.
TFZ × Z ⊂ R × R.
TFIf n is an Natural Number, the set {1,2,3,...,n} has n2 subsets.
T F For any set A, ∅ ⊂ P(A) and ∅ ∈ P(A).
T F The negation of the statement: All Natural Numbers are even
is the statement: Some Natural Numbers are not even.
T F [(36 DIV 5) − (93 MOD 7)] = 5.
TFIf d | (x + y), then d | x and d | y.
T F If A = {0,1}, then A × A × A = {000,001,010,011,100,101,110,111}.
TFIf Σ = {0,1}, then Σ5 = Σ × Σ × Σ × Σ × Σ.
T F The set of even integers and the set of odd integers partition the set of integers.
T F A conditional statement and its contrapositive are logically equivalent.
2. (6 pts.) Use the Euclidian Algorithm to find gcd(1000,60)
3. (10 pts.) Show, without using truth tables, that (~p ∧ q) → r ≡ ~p → (q → r).
4. (4 pts.) Give the converse, inverse, contrapositive, and negation of the universal state-
ment: All prime numbers greater than 2 are odd.
5. (10 pts.) Find the Disjunctive Normal Form of a circuit of four inputs in such a way
that if the integer value of the inputs is prime, then current flows out of the circuit. (For
example, 12 is not prime, and 12 = 1100, so f(1100) = 0)
6. (10 pts.) Show the following is a valid argument: p → (q ∧ r)
~r
∴ ~p
7. (40 pts.) Prove 2 of the 4 theorems:
Theorem 1: (A ∪ B) ∩ (A ∪ Cc) ∩ (Bc ∪ Cc) = (A − B) ∪ (B − C)
Theorem 2: For all integers a and b, if b is the successor of a, then b2 −a2 is odd.
Theorem 3: If every integer greater than 1 can be factored as the product of primes,then
there is no largest prime.
Theorem 4: If a, b, and c are integers with a = b + c , then gcd(a,b) = gcd(b,c).
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