Let X be a set with a binary operation and let a be in X. Assume...

 

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Let X be a set with a binary operation and let a be in X. Assume the operation has an identity element e. An element b is called a left inverse for a if ba = e. Similarly, it is a right inverse for a if ab = e. 1. In the set FA, find a simple, familiar condition for a function of to have a left inverse. If you have made a good answer, your condition will actually be necessary and sufficient. ('Necessary and sufficient condition' means that if the condition holds, then f has a left inverse and also that if f has a left inverse then the condition holds.) 2. Repeat for right inverses. 3. Give an example of a set A and function f which has a left inverse, but not a right inverse. Let X be a set with a binary operation and let a be in X. Assume the operation has an identity element e. An element b is called a left inverse for a if ba = e. Similarly, it is a right inverse for a if ab = e. 1. In the set FA, find a simple, familiar condition for a function of to have a left inverse. If you have made a good answer, your condition will actually be necessary and sufficient. ('Necessary and sufficient condition' means that if the condition holds, then f has a left inverse and also that if f has a left inverse then the condition holds.) 2. Repeat for right inverses. 3. Give an example of a set A and function f which has a left inverse, but not a right inverse.

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ANSWER 1 For a function f to have a left inverse in the set FA it must be true that fa e where e is ... View the full answer