# Identity Functions

1. Oct 15, 2008

### Gear300

From what I was reading, the apparent definition goes as: The Identity Function on E is the function IE from E into E defined by IE(x) = x. Since IE is the set of all ordered pairs (x,x) such that x ϵ E, IE is also called the diagonal subset of E x E.

If f is a function from E into F, clearly
1. f o IE = f,
2. IF o f = f, in which o is a composition operation

I understood 1., but I'm stuck on understanding on how 2. works. The definition is also confusing me; by how I read it, the identity function is the original function operating on itself...which, as stated, confuses me...any clarifications?

Last edited: Oct 15, 2008
2. Oct 15, 2008

### Pere Callahan

What is $(I_F\circ f)(x)$ for some element x of E?

3. Oct 15, 2008

### Gear300

In addition, could I get an example of an identity function for some arbitrary function (so I may further clarify my thoughts)?

4. Oct 15, 2008

### Pere Callahan

The identity function does not depend on any arbitrary function. It simply is the function $I_E:E\to E$ which returns its argument unchanged, that is $I_E(x)=x$ for all x in E. For any set E, there is exactly one such function.

5. Oct 15, 2008