CS 203 - Exam 1 - Spring 1996
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.
TFR ⊂ R×R.
T F The set {1,2,3,4} has 16 subsets.
TF∅ ⊂ P({∅,1}) and ∅ ∈ P({0,1}).
T F The negation of the statement: All Natural Numbers are positive
is the statement: No Natural Numbers are positive.
T F [(36 DIV 7) - (93 MOD 5)] = 2.
TFIf d | (x + y) and d | x, then 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| = 32.
T F The set of even integers and the set of odd integers partition the set of integers.
T F A conditional and its contrapositive are logically equivalent.
2. (6 pts.) Use the Euclidian Algorithm to find gcd(1234,56)
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 statement:
All prime numbers greater than 2 are odd.
5. (10 pts.) Find the Boolean polynomial representing 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 exam-
ple, 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)c = Ac ∪ Bc.
Theorem 2: The difference of the square of natural number and the square of its
successor is odd.
Theorem 3: There is no largest integer.
Theorem 4: If a, b, and c are integers with a = b + c , then gcd(a,b) = gcd(b,c).
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