# Let S be a set of three elements given by S = { A , B , C } . In the following table, all of the elements of S are listed in a row at the top and in a column at the left. The result x ∗ y is found in the row that starts with x at the left and in the column that has y at the top. For example, B ∗ C = C and C ∗ B = A . Thus the table defines the binary operation ∗ on the sets S . ∗ A B C A A B C B B C A C C A B 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?

### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

#### Solutions

Chapter
Section
Chapter 1.4, Problem 4E
Textbook Problem

## Let S be a set of three elements given by S = { A , B , C } . In the following table, all of the elements of S are listed in a row at the top and in a column at the left. The result x ∗ y is found in the row that starts with x at the left and in the column that has y at the top. For example, B ∗ C = C and C ∗ B = A . Thus the table defines the binary operation ∗ on the sets S . ∗ A B C A A B C B B C A C C A B 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?

Expert Solution

a)

To determine

Whether the given binary operation is commutative.

### Explanation of Solution

Given Information:

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

 ∗ A B C A A B C B B C A C C A B

Explanation:

Consider the given table

Expert Solution

b)

To determine

Whether there is an identity element in S.

Expert Solution

c)

To determine

The inverse of elements of S if there is any identity element.

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