infinitude of inverses
Let . Define Then, by induction, we see that . Next we want to show that if . Suppose and . Again by induction, we have
If we let then . So Equation 3 can be rewritten as . Then . Now, note that for , . This implies that
On the other hand, we also have
So combining the above two equations, we get . Let , then . Simplify, we have . Expanding , then
Then and we have reached a contradiction.
For the next part, notice that if and are two distinct right inverses of , then neither one of them can be a left inverse of , for if, say, , then . So we can apply the same technique used in the previous portion of the problem. Note that if , then
Multiply from the right, we have
Thus . Keep going until we reach , again a contradiction.
|Title||infinitude of inverses|
|Date of creation||2013-03-22 18:17:30|
|Last modified on||2013-03-22 18:17:30|
|Last modified by||CWoo (3771)|