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).