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Examination 1 - Spring 1995: Mathematics Problems, Exams of Discrete Structures and Graph Theory

A set of mathematical problems from an examination held in spring 1995. The problems cover various topics including logic, sets, and number theory. Students are required to use concepts such as power sets, boolean algebra, and the euclidean algorithm to solve the problems.

Typology: Exams

2012/2013

Uploaded on 04/27/2013

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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.
TFRZ = 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: pq
q~r
sr
ps
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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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.

T F R ∩ Z = N ∪ {0}.

T F.

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)}.

T F If Σ = {x,y,z} is an alphabet, then Σ^3 = {xxx, yyy, zzz}.

T F The set of positive integers and the set of negative integers partition the set of the integers.

T F (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.

  1. 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.

  1. 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)

  1. 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).

P A ( )

{ 1 2 4, , } ∈ P ( { 0 1 2 3 4 5 6 7 8 9, , , , , , , , , })

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