I was trying to work out the identity of a binary operation and I found that there were infinitely many possible identities for a certain value. My confusion is whether this is a valid identity or should there be a unique identity element?
Asked
Active
Viewed 365 times
1 Answers
3
It's unique. If $e$ and $e'$ are two identities ($\therefore$ left and right identities), then:
$e \star e' = e$ since $e'$ is a right identity.
$e\star e' = e'$ since $e$ is a left identity.
So $e = e'$.
-
the exact question is : (a, b)*(c, d) = (ac, bc + d), on the set {(x, y) ∈R x R} Now i get infintely many identity elements if i put a=0. So does an identity exist? – Dev Aggarwal Apr 18 '17 at 21:20
-
here's what i did - (a, b) * (e1, e2) = (a e1, b e1 + e2) = (a,b). Now if i put a=0, e1 can have any value.. – Dev Aggarwal Apr 18 '17 at 21:24
-
2An identity must work for every element, not just one. Consider over R: 2 * 0 = 0, but that doesn't mean 2 is an identity. – LtSten Apr 18 '17 at 21:26
-
Alright. Thanks for clearing that up – Dev Aggarwal Apr 18 '17 at 21:30