the cedar ledge

Images of Elements, Sets, and Functions

Date: October 15 2022

Summary: An overview on determining the image of elements, sets, and functions

Keywords: ##summary #image #set #function #element #subset #domain #range #archive

Bibliography

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Table of Contents

    1. Reading Motivation
    2. The Basics on the Images of Elements
      1. Example
    3. The Basics on the Images of Subsets
      1. Example
    4. The Basics on the Images of Functions
      1. Example
  1. How To Cite
  2. References:
  3. Discussion:

Reading Motivation

I wanted to understand more about the ideas of images as an avenue to understand bundles.

The Basics on the Images of Elements

If there is some element xXx \in X, and some function ff, the image of xx under ff (i.e. the image of the element), is the value of ff applied to xx:

im(x)=f(x) im(x) = f(x)

In other words, I would read this as

Given some element xx, its image is whatever ff maps xx to – the function's output for xx.

Example

Let x=4x = 4 and f:xxf : x \rightarrow \sqrt{x}. What is the image of xx?

Using the definition:

im(x)=f(x) im(x) = f(x)

We can then know:

im(x)=f(4)=2 im(x) = f(4) = 2

The Basics on the Images of Subsets

Given f:XYf : X \rightarrow Y and WXW \subseteq X, the image of WW under ff is:

im(f(W))=f(W)={f(x)xW} im(f(W)) = f(W) = \{f(x) | x \in W \}

How I might read this is:

If we have two sets, XX and YY, and a subset WXW \subseteq X (a subset of XX) while also having a function that maps values of XX to YY and we want to know the image of the subset, WW, we can have ff operate on each element xWx \in W. This generates the image of WW!

Example

Let X=RX = \mathbb R, Y=+RY = +\mathbb R, W={1,2,3,4,5}W = \{1, 2, 3, 4, 5\} and f:xx2f : x \rightarrow x^{2}. What is the image of WW under ff?

Using the generic definition:

im(f(W))=f(W)={f(x)xW} im(f(W)) = f(W) = \{f(x) | x \in W \}

We can then write the set as follows:

im(f(W))={1,4,9,16,25} im(f(W)) = \{1, 4, 9, 16, 25\}

The Basics on the Images of Functions

If we have two sets, XX and YY, and a function f:XYf : X \rightarrow Y which sends xXx \in X to YY, the image of this function (denoted im(f)im(f)), is:

im(f)={f(x)xX}Y im(f) = \{f(x) | x \in X\} \subseteq Y

So, what does that mean or how do we read this? In short, I would read this as:

Given xx as an element in XX, the image of the function ff are the values that xx is mapped to in YY via ff. Because each value is mapped to the domain YY, those values form a subset of YY.

This explanation may look familiar as the image of a function is actually synonymous to a function's range! 😱 Typically though, to be a bit more rigorous, the word range is not used in this setting but rather image.

Example

Let X=RX = \mathbb R, Y=+RY = +\mathbb R, and f:xx2f : x \rightarrow x^{2}. What is the image of ff?

Using the generic definition:

im(f)={f(x)xX}Y im(f) = \{f(x) | x \in X\} \subseteq Y

We can then write the set as follows:

im(f)={0,...,.01,...,25,...,}Y im(f) = \{0, ..., .01, ..., 25, ..., \infin\} \subseteq Y

In this example, based on the definition we have for an im(f)im(f), the im(f)im(f) is all possible outputs that ff can generate which is the range +R+\mathbb R.

How To Cite

Zelko, Jacob. Images of Elements, Sets, and Functions. https://jacobzelko.com/10152022173643-image-of-sets. October 15 2022.

References:

Discussion:

CC BY-SA 4.0 Jacob Zelko. Last modified: January 17, 2023. Website built with Franklin.jl and the Julia programming language.