the cedar ledge

# The Preimage of Elements, Sets, and Functions

Date: October 28 2022

Summary: How the preimage appears in elements, sets, and functions

Keywords: #archive #sets #preimage #elements #functions #inverse #image

# Bibliography

Not Available

As I have learned and understood what images are, I am now on my quest to learn what preimages so that I can understand the concept of bundles!

### Basics on the Preimage of a Subset

Given some function $f : A \rightarrow B$ and the subsets $C \subseteq A$ and $D \subseteq B$, then the preimage (or inverse image of a set) is:

$f^{-1}(D) = \{x \in A : f(x) \in D\}$

Which has the interesting property that the image of each element in $D$ is in $C$.

For some reason, admittedly, this was more challenging for me to understand than the image of a subset. Why that is, I do not know; maybe my brain has a bit of a challenge thinking in reverse? Personally, I find imagining the notation of the definition more easily followed when it is written like this:

$f^{-1}(D) = \{x \in A : f(x) = D\}$

Notice the subtle notation change I introduced where I said $=$ instead of $\in$ which, although interchangeable here, reads more clearly in my mind. Essentially, you say the condition for this set to be any element, $x$, that satisfies the equation, $f(x) = D$. To me, my brain translates that condition into a fun challenge rather than an abstract $\in D$ which reads less concretely.

#### Example

Let $X = \R$, $Y = \R$, $V = \{1, 4, 9, 16, 25\}$ which we know the fact that $V \subseteq Y$, and $f : x \rightarrow x^{2}$.

Using the definition:

$f^{-1}(V) = \{x \in X : f(x) \in V\}$

We can create a few equations that can help us generating the set:

$x^{2} = 1$ $x^{2} = 4$ $x^{2} = 9$ $x^{2} = 16$ $x^{2} = 25$

And solving these equations gives the following solutions:

$x^{2} = 1; x = \pm 1$ $x^{2} = 4; x = \pm 2$ $x^{2} = 9; x = \pm 3$ $x^{2} = 16; x = \pm 4$ $x^{2} = 25; x = \pm 5$

And we can write our solution to the inverse image of the set $V$ being the following:

$\{-5, -4, -3, -2, -1, +1, +2, +3, +4, +5\}$

## How To Cite

Zelko, Jacob. The Preimage of Elements, Sets, and Functions. https://jacobzelko.com/10282022132046-preimage-of-sets. October 28 2022.

## Discussion:

CC BY-SA 4.0 Jacob Zelko. Last modified: May 19, 2024. Website built with Franklin.jl and the Julia programming language.