Explanation Construct a mathematical system with three elements that is { E , A , B } under the operation ‘ Φ ’. Here, Φ defined as follows. Φ E A B E E A B A A B E B B E A Consider the above three element system with operation Φ is a set. Now, check for the system to be a group. Closure: Let A , B ∈ { E , A , B } . Observe that, A Φ B is E that also belongs to the set { E , A , B } . Thus, closure property satisfies in the binary operation of ‘ Φ ’. Identity: From given, the identity element is E . From the given table, it is observed that A Φ E = A , B Φ E = B and E Φ E = E . Hence, E is the identity element in the set { E , A , B } . Thus, identity property satisfies in the binary operation of ‘ Φ ’. Inverse: Let a ∈ { E , A , B } , then, a Φ ( − a ) = ( − a ) Φ a = 0 . This gives − a as the inverse of a , that shows, for every element of set, the inverse exists. If an element added with its inverse, it should give the identity element under an operation ‘ Φ ’. From table it is observed that, A Φ B = E , hence B is the inverse of A, B Φ A = E , hence A is the inverse of B and E Φ E = E , C is its own inverse. Associative property: Let a , b , c ∈ { E , A , B } , then ( a Φ b ) Φ c = a Φ ( b Φ c )

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 9.2, Problem 51E

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To explain: Whether the mathematical system of the set of 3 elements is a commutative group.

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Chapter 9.2, Problem 50E

Chapter 9.2, Problem 52E

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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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Whether the mathematical system of the set of 3 elements is a commutative group.

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