   Chapter 1.4, Problem 6E

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# Let S be the set of four elements given by S = { A , B , C , D } with the following table. ∗ A B C D A A A A A B A B A B C A A C C D A B C D a. Is the binary operation ∗ commutative? Why?b. Determine whether there is an identity element in S for ∗ .c. If there is an identity element, which elements have inverses?

(a)

To determine

Whether the binary operation is commutative on set S={A,B,C,D}.

Explanation

Given Information: The set S={A,B,C,D} is given and the table that defines the binary operation on the set S is given as:

 ∗ A B C D A A A A A B A B A B C A A C C D A B C D

Formula Used:

A binary operation is said to be commutative for any elements a,bA if ab=ba.

Explanation: Consider the given table

(b)

To determine

Whether there is an identity element in S.

(c)

To determine

Whether there is an inverse element of S.

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