Examination 1 - Spring 1995
Symbols: N denotes the Natural Numbers, Z denotes the Integers, Q denotes the Rational Numbers,
and R denotes the Real Numbers, 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 = N ∪ {0}.
TF .
T F The contrapositive of the statement I finished my work implies I could go 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
q → ~r
s → r
p ∧ s
T F If A = {x,y,z}, then A × A = {(x,x),(x,y),(x,z),(y,x),(y,y),(y,z),(z,x),(z,y),(z,z)}.
TFIf Σ = {x,y,z} is an alphabet, then Σ3 = {xxx, yyy, zzz}.
T F The set of positive integers and the set of negative integers partition the set of the integers.
TF(Show work!) [[(33 mod 12) - (154 div 100)]] mod 7 = 1 .
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 If t represents tautology, then (p ∧ t) ≡ p and (p ∨ t) ≡ t.
2. Use the Euclidean Algorithm to find gcd(10240,3200).
3. Show, without using truth tables, that p ∨ (p ∧ q) ≡ p. (HINT: Is any part of #1 helpful?)
4. Write the negation of the existential statement: There exists x ∈ N such that x is even and x is prime.
5. Find the Boolean polynomial representing a circuit of three switches controlling a light bulb in such a
way that if the middle switch is the opposite of the first and third, then the bulb turns on.
6. For the sets Y = {x,y}, A = {1,2,3}, and B = {2,3,4}, show that (B − C) × Y = (B × Y) − (C × Y)
7. Prove 2 of the 4 theorems:
Theorem 1: Given that each positive integer has a unique representation as the product of prime
numbers, then the set of prime numbers is infinite.
Theorem 2: If a and b are positive integers such that a|b and b|a, then a = b.
Theorem 3: If Y, B, and C are sets, then Y × (B − C) = (Y × B) − (Y × C).
Theorem 4: If a,b,q, and r are positive integers with a = bq + r and 0< r < b, then gcd(a,b) = gcd(b,r).
PA()
124,,{}P0123456789,,,,,,,,,{}()∈
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