GROUP holds four properties simultaneously: i) Closure, ii) Associative, iii) Identity element; and iv) Inverse element. Give an illustrative example for each statement to prove whether GROUP or NOT A GROUP. 1. The set of integers Z, the set of rational numbers Q, and the set of real numbers R are all groups under ordinary addition. 2. The set of integers under ordinary multiplication is not a group. 3. The set of positive irrational numbers together with 1 under multiplication satisfies the three properties given in the definition of a group but is not a group. Why? 4. The set R* of nonzero real numbers is a group under ordinary multiplication. 5. A rectangular array of the form [] is called a 2 x 2 matrix. The set of all 2 x 2 matrices with real entries is a group under component wise addition.

GROUP holds four properties simultaneously: i) Closure, ii) Associative, iii) Identity element; and iv) Inverse element. Give an illustrative example for each statement to prove whether GROUP or NOT A GROUP. 1. The set of integers Z, the set of rational numbers Q, and the set of real numbers R are all groups under ordinary addition. 2. The set of integers under ordinary multiplication is not a group. 3. The set of positive irrational numbers together with 1 under multiplication satisfies the three properties given in the definition of a group but is not a group. Why? 4. The set R* of nonzero real numbers is a group under ordinary multiplication. 5. A rectangular array of the form [] is called a 2 x 2 matrix. The set of all 2 x 2 matrices with real entries is a group under component wise addition.

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter2: Working With Real Numbers

Section2.1: Basic Assumptions

Problem 40WE

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