Identity Function Clarification: Definition & Examples

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Discussion Overview

The discussion centers around the definition and properties of the identity function, particularly in the context of function composition. Participants explore the implications of the identity function on other functions and seek clarification on its definition and examples.

Discussion Character

  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant defines the identity function on a set E as IE(x) = x, noting that it is represented by the diagonal subset of E x E.
  • The same participant expresses confusion about the operation IF o f = f, seeking clarification on how this composition works.
  • Another participant asks for the expression (I_F ∘ f)(x) for an element x of E, indicating a desire to understand the application of the identity function in composition.
  • A later reply asserts that the identity function is independent of any arbitrary function and emphasizes that it simply returns its argument unchanged.
  • Participants express varying levels of understanding and seek examples to further clarify their thoughts on the identity function.

Areas of Agreement / Disagreement

Participants generally agree on the definition of the identity function, but there is some confusion regarding its application in function composition. The discussion remains unresolved regarding the specific implications of the identity function in various contexts.

Contextual Notes

Participants have not fully resolved the implications of the identity function in function composition, and there are varying interpretations of its relationship with other functions.

Gear300
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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:
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What is [itex](I_F\circ f)(x)[/itex] for some element x of E?
 
Pere Callahan said:
What is [itex](I_F\circ f)(x)[/itex] for some element x of E?

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)?
 
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.
 
I see...your answer clarifies things for me...thanks.
 

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