# identity function

#### 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 $$\displaystyle f ^{-1}\circ f=1$$

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

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

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#### Plato

MHF Helper
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 $$\displaystyle f \circ f^{-1}:B\to B$$.

• novice

#### novice

Are you sure that you have not turned things around here.
For one, if $$\displaystyle f:A\to B$$ then $$\displaystyle f \circ f^{-1}:B\to B$$.

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

#### Plato

MHF Helper
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 $$\displaystyle f^{-1}\circ f(x)=1_A(x)=x$$.

• novice

#### novice

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.