Identity Function Clarification: Definition & Examples

In summary, the conversation discusses the definition and properties of the identity function on a set E, denoted as IE, which is defined as IE(x) = x for all x in E. It is also known as the diagonal subset of E x E. The conversation also mentions the composition operation, with the property that f o IE = f and I_F o f = f. The conversation also includes a request for an example of an identity function for an arbitrary function, to which the response is that the identity function is unique for any set E and is defined as I_E(x) = x for all x in E.
  • #1
Gear300
1,213
9
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?
 
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  • #2
What is [itex](I_F\circ f)(x)[/itex] for some element x of E?
 
  • #3
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)?
 
  • #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.
 
  • #5
I see...your answer clarifies things for me...thanks.
 

FAQ: Identity Function Clarification: Definition & Examples

What is an identity function?

An identity function is a mathematical concept that maps every element from a set to itself. In other words, it is a function that returns the same value that is inputted.

How is an identity function written?

An identity function is typically written as f(x) = x, where f represents the function and x represents the input value. This notation is used to show that the output of the function is equal to the input value.

What are some examples of identity functions?

Some common examples of identity functions include the identity function in algebra, which is f(x) = x, the identity function in calculus, which is f(x) = x, and the identity function in set theory, which is f(x) = x.

How is an identity function useful?

An identity function may seem simple, but it is a fundamental concept in mathematics and has many practical applications. It is used to prove theorems, simplify equations, and serve as a building block for more complex functions.

Is an identity function the same as an inverse function?

No, an identity function is not the same as an inverse function. While an identity function returns the input value, an inverse function performs the opposite operation and returns the original input value. In other words, an inverse function "undoes" the original function.

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