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Identity Functions

  1. Oct 15, 2008 #1
    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. jcsd
  3. Oct 15, 2008 #2
    What is [itex](I_F\circ f)(x)[/itex] for some element x of E?
  4. Oct 15, 2008 #3
    oh...I sort of understand now...interesting...your presentation suddenly made sense to me...thanks.

    In addition, could I get an example of an identity function for some arbitrary function (so I may further clarify my thoughts)?
  5. Oct 15, 2008 #4
    The identity function does not depend on any arbitrary function. It simply is the function [itex]I_E:E\to E[/itex] which returns its argument unchanged, that is [itex]I_E(x)=x[/itex] for all x in E. For any set E, there is exactly one such function.
  6. Oct 15, 2008 #5
    I see...your answer clarifies things for me...thanks.
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