# What is the notation $id_{ }$?

1. Jul 23, 2012

### operationsres

Preface: Can't understand notation from 1st year Real Analysis textbook that they pulled out of thin air without defining.

1. The thing statement that they're proving
Let A and B be sets. Moreover, let $f : A \rightarrow B$ be some injective map. Calculate $f^{-1} \circ f$.

SOLUTION: To every $y \in f(A)$, the map $f^{-1}$ associates the corresponding $x \in A$ which satisfies $f(x) = y$. In particular, it associates with $f(x)$ the element $x$ for all $x \in A$. Hence:

$f^{-1} \circ f = id_A$, $f \circ f^{-1} = id_{f(A)}$,

where for every set $C$ the corresponding map $id_C : C \rightarrow C$ is defined by:

$id_C (x) := C$

for all $x \in C$.

2. My confusion
What's $id$?

2. Jul 23, 2012

### HallsofIvy

Staff Emeritus
The identity map- the function that maps every member of a set to itself. If you are taking "mathematical analysis" surely you have seen inverse functions before? And you should know that the definition of "inverse function" is that f-1(f(x))= f(f-1(x))= x?

But I doubt that your text says "idC(x):= C". Surely it says "idC(x)= x for all $x\in C$". Check that again.

3. Jul 23, 2012

### operationsres

1) It does say what you supposed that it did not say. Must be a typo.

Page 31 of Beyer's Calculus & Analysis: A combined approach.

2) You've solve my problem. Thanks :)