Preimage theorem

Imagine you have a big jigsaw puzzle with many different pieces. You want to take one of the completed puzzle images and figure out which group of puzzle pieces made that image. This is kind of like finding a "preimage".

In math, we often use functions that take in some input (like puzzle pieces) and give us an output (like a completed puzzle image). The preimage theorem says that if we have a function and we know what the output is, we can always find the group of inputs that make that output.

So, going back to our puzzle example, if we have a function that takes in the shape of the puzzle pieces and gives us the completed puzzle image as output, then we can look at the completed puzzle image and figure out which group of puzzle pieces made it.

The preimage theorem is very helpful in math because it allows us to work backwards from an output to find the inputs that make it. This can be useful in many different situations, like cryptography or computer science.