Explanation The set of all integers under the operation of addition can be a group. The properties of a group as follows, (i) The set of elements is closed under the operation. (ii) An identity element exists for the set under the operation. (iii) Every element in the set has an inverse under the operation. (iv) The set of elements is associative under the operation. Example: (i) Consider a = 1 and b = 2 in the set of integers ℤ . Check whether considered integers are closed or not. a + b = 1 + 2 = 3 ∈ ℤ Here, the sum of the elements in the set. That is, the closure property holds. (ii) Consider a = 5 in the set of integers ℤ . Check whether 0 is additive identity or not. a + 0 = 5 + 0 = 5 ∈ ℤ Therefore, the sum of 0 and 5 (integer) is 5 and 0 is the additive identity element under addition. That is, identity property holds. (iii) Consider a = 4 in the set of integers ℤ . Check whether 0 is inverse or not. a + ( − a ) = 4 + ( − 4 ) = 4 − 4 = 0 ∈ ℤ Therefore, the sum of 4 (integer) and it negative is 0 is an inverse element under addition

A Survey of Mathematics with Applications (10th Edition) - Standalone book

10th Edition

ISBN: 9780134112107

Author: Allen R. Angel, Christine D. Abbott, Dennis Runde

Publisher: PEARSON

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Chapter 9.1, Problem 8E

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To fill: If a mathematical system possesses the following properties-closure, identity element, inverses, and the associative property-then the associative property-then the mathematical system is a( n ) ________.

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Chapter 9.1, Problem 7E

Chapter 9.1, Problem 9E

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A Survey of Mathematics with Applications (10th Edition) - Standalone book

Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 10E

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If a mathematical system possesses the following properties-closure, identity element, inverses, and the associative property-then the associative property-then the mathematical system is a( n ) ________.

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To fill: If a mathematical system possesses the following properties-closure, identity element, inverses, and the associative property-then the associative property-then the mathematical system is a( n ) ________.

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