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 theset {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 notationand 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 measuresimilarity 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 thatproperty. [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?

Given system:.To find:a) Name the systems, which satisfies the following operations.i) ii) iii)…

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.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.2: Determinants

Problem 19AEXP

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