Date: October 28 2022
Summary: How the preimage appears in elements, sets, and functions
Keywords: #archive #sets #preimage #elements #functions
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!
Given some function and the subsets and , then the preimage (or inverse image of a set) is:
Which has the interesting property that the image of each element in is in .
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:
Notice the subtle notation change I introduced where I said instead of which, although interchangeable here, reads more clearly in my mind. Essentially, you say the condition for this set to be any element, , that satisfies the equation, . To me, my brain translates that condition into a fun challenge rather than an abstract which reads less concretely.
Let , , which we know the fact that , and .
Using the definition:
We can create a few equations that can help us generating the set:
And solving these equations gives the following solutions:
And we can write our solution to the inverse image of the set being the following:
Zelko, Jacob. The Preimage of Elements, Sets, and Functions. https://jacobzelko.com/10282022132046-preimage-of-sets. October 28 2022.