   Chapter 1.4, Problem 16E

Chapter
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Textbook Problem
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# Assume that ∗ is an associative binary operation on A with an identity element. Prove that the inverse of an element is unique when it exists.

To determine

To prove: If A contains an identity element with respect to , then the inverse of an element is unique when it exists, where be a binary operation on the non empty set A.

Explanation

Given Information:

be a binary operation on the non empty set A.

Formula Used:

If e is the identity element of A and aA then, ea=ae=a and if a1 is the inverse of a then aa1=e.

Explanation:

Let xA be an element whose inverse exists and let y,zA are the inverses of x.

This means xy=yx=e and xz=zx=e, where e is an identity of A

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