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

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Table of Contents

    1. Reading Motivation
    2. Basics on the Preimage of a Subset
      1. Example
  1. How To Cite
  2. References:
  3. Discussion:

Reading Motivation

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:ABf : A \rightarrow B and the subsets CAC \subseteq A and DBD \subseteq B, then the preimage (or inverse image of a set) is:

f1(D)={xA:f(x)D} f^{-1}(D) = \{x \in A : f(x) \in D\}

Which has the interesting property that the image of each element in DD is in CC.

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:

f1(D)={xA:f(x)=D} 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, xx, that satisfies the equation, f(x)=Df(x) = D. To me, my brain translates that condition into a fun challenge rather than an abstract D\in D which reads less concretely.

Example

Let X=RX = \mathbb R, Y=RY = \mathbb R, V={1,4,9,16,25}V = \{1, 4, 9, 16, 25\} which we know the fact that VYV \subseteq Y, and f:xx2f : x \rightarrow x^{2}.

Using the definition:

f1(V)={xX:f(x)V} f^{-1}(V) = \{x \in X : f(x) \in V\}

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

x2=1 x^{2} = 1 x2=4 x^{2} = 4 x2=9 x^{2} = 9 x2=16 x^{2} = 16 x2=25 x^{2} = 25

And solving these equations gives the following solutions:

x2=1;x=±1 x^{2} = 1; x = \pm 1 x2=4;x=±2 x^{2} = 4; x = \pm 2 x2=9;x=±3 x^{2} = 9; x = \pm 3 x2=16;x=±4 x^{2} = 16; x = \pm 4 x2=25;x=±5 x^{2} = 25; x = \pm 5

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

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

References:

Discussion:

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