What is the notation [itex]id_{ }[/itex]?

[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?

 

[HallsofIvy]

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 "id_C(x):= C". Surely it says "id_C(x)= x for all x\in C". Check that again.

 

[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 :)