Let Z denote the set of integers. If m is a positive integer, we write Zm for the system "integers modulo m." Some authors write Z/mZ for that system. For completeness, we include some definitions here. The system Zm can be represented as the set {0, 1,..., m - 1} with operations (addition) and (multiplication) defined as follows. If a, b are elements of {0, 1,..., m - 1}, define: a + b = the element c of {0, 1,..., m - 1} such that a+b-c is an integer multiple of m. a ob = the element d of {0, 1,..., m - 1} such that ab -d is an integer multiple of m.
Let Z denote the set of integers. If m is a positive integer, we write Zm for the system "integers modulo m." Some authors write Z/mZ for that system. For completeness, we include some definitions here. The system Zm can be represented as the set {0, 1,..., m - 1} with operations (addition) and (multiplication) defined as follows. If a, b are elements of {0, 1,..., m - 1}, define: a + b = the element c of {0, 1,..., m - 1} such that a+b-c is an integer multiple of m. a ob = the element d of {0, 1,..., m - 1} such that ab -d is an integer multiple of m.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let Z denote the set of integers. If m is a positive integer, we write Zm for the system of
"integers modulo m." Some authors write Z/mZ for that system.
For completeness, we include some definitions here. The system Zm can be represented as the
set {0, 1,..., m - 1} with operations (addition) and (multiplication) defined as follows.
If a, b are elements of {0, 1,..., m - 1}, define:
ab the element c of {0, 1,...,m - 1} such that a +b-c is an integer multiple of m.
a b = the element d of {0, 1,..., m - 1} such that ab -d is an integer multiple of m.
For example, 30 4 = 2 in Z5, 303= 1 in Z4, and -1 = 12 in Z₁3.
To simplify notations (at the expense of possible confusion), we abandon that new notation
and write a + b and ab for the operations in Zm, rather than writing ab and a b.
=
Let Q denote the system of rational numbers.
We write 4Z for the set of multiples of 4 in Z. Similarly for 4Z12.
Consider the following number systems:
Z, Q, 4Z, Z3, Z8, Z9, 4Z12, Z13.
One system may be viewed as similar to another in several different ways. We will measure
similarity using only algebraic properties.
(a) Consider the following sample properties:
(i) If a² = 1, then a = ±1.
(ii) If 2x = 0, then x = 0.
(iii) If c² = 0, then c = 0.
Which of the systems above have properties (i), (ii), and/or (iii)?
(b) Formulate another algebraic property and determine which of those systems have that
property. [Note: Cardinality is not considered to be an algebraic property.]
Write down some additional algebraic properties and investigate them.
(c) In your opinion, which of the listed systems are "most similar" to each another?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6aa63a3-598f-4809-8e47-c6f7fb9d3551%2F7677be3b-dcca-49d4-aee2-f7755be844e7%2F4v7tqny_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let Z denote the set of integers. If m is a positive integer, we write Zm for the system of
"integers modulo m." Some authors write Z/mZ for that system.
For completeness, we include some definitions here. The system Zm can be represented as the
set {0, 1,..., m - 1} with operations (addition) and (multiplication) defined as follows.
If a, b are elements of {0, 1,..., m - 1}, define:
ab the element c of {0, 1,...,m - 1} such that a +b-c is an integer multiple of m.
a b = the element d of {0, 1,..., m - 1} such that ab -d is an integer multiple of m.
For example, 30 4 = 2 in Z5, 303= 1 in Z4, and -1 = 12 in Z₁3.
To simplify notations (at the expense of possible confusion), we abandon that new notation
and write a + b and ab for the operations in Zm, rather than writing ab and a b.
=
Let Q denote the system of rational numbers.
We write 4Z for the set of multiples of 4 in Z. Similarly for 4Z12.
Consider the following number systems:
Z, Q, 4Z, Z3, Z8, Z9, 4Z12, Z13.
One system may be viewed as similar to another in several different ways. We will measure
similarity using only algebraic properties.
(a) Consider the following sample properties:
(i) If a² = 1, then a = ±1.
(ii) If 2x = 0, then x = 0.
(iii) If c² = 0, then c = 0.
Which of the systems above have properties (i), (ii), and/or (iii)?
(b) Formulate another algebraic property and determine which of those systems have that
property. [Note: Cardinality is not considered to be an algebraic property.]
Write down some additional algebraic properties and investigate them.
(c) In your opinion, which of the listed systems are "most similar" to each another?
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