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

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 $x \in X$, and some function $f$, the image of $x$ under $f$ (i.e. the image of the element), is the value of $f$ applied to $x$:

In other words, I would read this as

Given some element $x$, its image is whatever $f$ maps $x$ to – the function's output for $x$.

Example

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

Using the definition:

We can then know:

The Basics on the Images of Subsets

Given $f : X \rightarrow Y$ and $W \subseteq X$, the image of $W$ under $f$ is:

How I might read this is:

If we have two sets, $X$ and $Y$, and a subset $W \subseteq X$ (a subset of $X$) while also having a function that maps values of $X$ to $Y$ and we want to know the image of the subset, $W$, we can have $f$ operate on each element $x \in W$. This generates the image of $W$!

Example

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

Using the generic definition:

We can then write the set as follows:

The Basics on the Images of Functions

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

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

Given $x$ as an element in $X$, the image of the function $f$ are the values that $x$ is mapped to in $Y$ via $f$. Because each value is mapped to the domain $Y$, those values form a subset of $Y$.

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 $X = \R$, $Y = +\R$, and $f : x \rightarrow x^{2}$. What is the image of $f$?

Using the generic definition:

We can then write the set as follows:

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

How To Cite

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

References:

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