In Example 8.3.12, define operations of addition (+) and multiplication (·) as follows: For every
a. Prove that this addition is well defined. That is, show that if
b. Prove that this multiplication is well defined. That is, show that if
c. Show that [(0, 1)] is an identity element for addition. That is, show that for any
d. Find an identity element for multiplication. That is, find (i, j) in A so that for every (a, b) in
e. For any
f. Given any
identity element you found in part (d).
(a)
To show that if
Given information:
The operations of addition (+)and multiplication (·) as follows: For all
Proof:
Let us assume that
In general:
By the definition of the equivalence class:
(b)
To prove:
If
(c)
To show that
(d)
An identity element for multiplication for the given operation. That means, find
(e)
To prove:
To show that
(f)
An inverse for