# In Example 8.3.12, define operations of addition (+) and multiplication (·) as follows: For every ( a , b ) , ( c , d ) ∈ A , [ ( a , b ) ] + [ ( c , d ) ] = [ ( a d + b c , ​ b d ) ] [ ( a , b ) ] ⋅ [ ( c , d ) ] = [ ( a c , b d ) ] . a. Prove that this addition is well defined. That is, show that if [ ( a , b ) ] = [ ( a ' , b ' ) ] and [ ( c , d ) ] = [ ( c ' , d ' ) ] , then [ ( a d + b c , b d ) ] = [ ( a ' d ' + b ' c ' , b ' d ' ) ] . b. Prove that this multiplication is well defined. That is, show that if [ ( a , b ) ] = [ ( a ' , b ' ) ] and [ ( c , d ) ] = [ ( c ' , d ' ) ] . then [ ( a c + b d ) ] = [ ( a ' c ' , b ' d ' ) ] . c. Show that [(0, 1)] is an identity element for addition. That is, show that for any ( a , b ) ∈ A , [ ( a , b ) ] + [ ( 0 , 1 ) ] = [ ( 0 , 1 ) ] + [ ( a , b ) ] = [ ( a , b ) ] . d. Find an identity element for multiplication. That is, find ( i , j ) in A so that for every ( a , b ) in A , [ ( a , ​ b ) ] ⋅ [ ( i , j ) ] = [ ( i , j ) ] ⋅ [ ( a , ​ b ) ] = [ ( a , ​ b ) ] . e. For any ( a , b ) ∈ A , show that [(- a , b )] is an inverse for [ ( a , ​ b ) ] for addition. That is, show that [ ( − a , b ) ] + [ ( a , b ) ] = [ ( a , b ) ] + [ ( − a , b ) ] = [ ( 0 , 1 ) ] . f. Given any ( a , b ) ∈ A with a ≠ 0 , find an inverse for [ ( a , ​ b ) ] for multiplication. That is, find ( c, d ) in A so that [ ( a , b ) ] ⋅ [ ( c , d ) ] = [ ( c , d ) ] ⋅ [ ( a , b ) ] = [ ( i , j ) ] . where [ ( i , j ) ] is the identity element you found in part (d).

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

#### Solutions

Chapter
Section
Chapter 8.3, Problem 43ES
Textbook Problem

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