# Thread: identity function

1. ## identity function

I found the following from a book with no proof:

Let $\displaystyle f:A \rightarrow A$ be defined by the formula $\displaystyle f(x)=x$, then $\displaystyle f$ is called the identity function, denoted by $\displaystyle 1$ or by $\displaystyle 1_A$.

Let $\displaystyle f:A \rightarrow B$ and it has the inverse function $\displaystyle f^{-1}:B\rightarrow A$, then $\displaystyle f ^{-1}\circ f=1$

Question: Is it true that $\displaystyle f ^{-1}\circ f=1$?

Isn't $\displaystyle f(x)^{-1} \circ f(x)=x$?

2. Originally Posted by novice
I found the following from a book with no proof:

Let $\displaystyle f:A \rightarrow A$ be defined by the formula $\displaystyle f(x)=x$, then $\displaystyle f$ is called the identity function, denoted by $\displaystyle 1$ or by $\displaystyle 1_A$.

Let $\displaystyle f:A \rightarrow B$ and it has the inverse function $\displaystyle f^{-1}:B\rightarrow A$, then $\displaystyle f \circ f^{-1}=1$

Question: Is it true that $\displaystyle f \circ f^{-1}=1$?

Isn't $\displaystyle f(x) \circ f^{-1}(x)=x$?
Are you sure that you have not turned things around here.
For one, if $\displaystyle f:A\to B$ then $\displaystyle f \circ f^{-1}:B\to B$.
So what are you asking?

3. Originally Posted by Plato
Are you sure that you have not turned things around here.
For one, if $\displaystyle f:A\to B$ then $\displaystyle f \circ f^{-1}:B\to B$.
So what are you asking?
Yes, made a mistake. Sorry.

Question: Is it true that $\displaystyle f^{-1}\circ f =1$?

4. But it is just a matter of notation: $\displaystyle f\circ f^{-1}=1_B~\&~ f^{-1}\circ f=1_A$.
Recall that $\displaystyle \left( {\forall x \in A} \right)\left[ {1_A (x) = x} \right]$ thus $\displaystyle f^{-1}\circ f(x)=1_A(x)=x$.

5. Originally Posted by Plato
But it is just a matter of notation: $\displaystyle f\circ f^{-1}=1_B~\&~ f^{-1}\circ f=1_A$.
Recall that $\displaystyle \left( {\forall x \in A} \right)\left[ {1_A (x) = x} \right]$ thus $\displaystyle f^{-1}\circ f(x)=1_A(x)=x$.
I found $\displaystyle i_A$ a much better notation. Perhaps, the book is too old. The inverse function denoted by 1 is terribly confusing. I have mistaken it as a number.