1. ## Unique identity

Suppose that * is a binary operation on a non-empty set S. Show that if (*,S) contains an identity e then e is unique.

Is this right?

e1 * x = x * e1 = x

x = e2

e1 * e2 = e2 * e1 = e2

then do for e2

e2 * x = x * e2 = x

x = e1

e2 * e1 = e1 * e2 = e1

Suppose that * is a binary operation on a non-empty set S. Show that if (*,S) contains an identity e then e is unique.

Is this right?

e1 * x = x * e1 = x

x = e2

e1 * e2 = e2 * e1 = e2

then do for e2

e2 * x = x * e2 = x

x = e1

e2 * e1 = e1 * e2 = e1
I would say "let x= e2" and "let x= e1" (so a reader won't think you are saying that a previously defined x is equal to e2) but other than that, this is fine.

3. Originally Posted by HallsofIvy
I would say "let x= e2" and "let x= e1" (so a reader won't think you are saying that a previously defined x is equal to e2) but other than that, this is fine.
thanks, this site is brilliant.

Ill contribute after my exams are over and done with

4. I think a tad shorter is to note that $e=ee'=e'$.