Imagine the two-dimensional plane, something you probably saw when you were in highschool math. Now, if you needed to describe a point on that plane, a natural way to do so would be to provide two numbers; one for the horizontal distance from the origin, and one for the vertical distance. Well would you look at that, you just gave me a vector!! A vector is just a mathematical object (like a number or variable) that has 2 elements. Visually, you can imagine a vector as an arrow that points from (0,0) to the location it represents. They are super useful for representing all kinds of things, and have a lot of interesting structure underneath the hood. Below is a simple example of a vector:

A matrix can feel daunting the first time you see one. Whereas a vector is a "list" of numbers (i.e. [5,1,3] is a vector), a matrix is a 2D array of numbers! Here's a visual example of what I mean if that doesn't quite make sense to you:

If you kinda squint your eyes, you can think about a matrix like a bunch of vectors smushed together. And well, yeah, they kinda are. You can always break a matrix into a combination of a bunch of vectors, so here's an analogy: In terms of complexity, If vectors are addition, matrices are multiplication.

Now you may be wondering, what this could possibly represent in applied areas? Surely this is just some wacky abstract object mathematicians made up for no good reason. Well, you'd be dead WRONG my friend, as matrices are some of the most powerful tools for solving complex systems, describing how things change over time, and all sorts of other cool stuff. Matrices are freaking epic, basically. But... they're also kinda complicated. I may addend this section with a more in depth exploration of matrices later, but understanding their relationship to vectors is sufficient for now.

Sets can feel daunting when you see capital S Set Theory, but really they are very simple. A set is just a collection of objects (usually called elements). No really, thats all a set is. For example, the set {1,3,5,7,...} is just a group of all the odd-valued natural numbers. the set {apple, banana, mango, peach} is the set of my favorite fruit. Notice how the elements of a set don't have to be numbers? Kinda cool. There are lots of ways to define a set, but usually we give some rule called the generator that acts like a machine to generate elements given some logical expression.

A quick example of a set generator would be B = { e | e<5 }. This reads, "B is the set of all elements e such that e is less than 5." And boom! We made our very own set, and an infinite one at that! Now what if I thought about the set A = { i | i<4 } ("A is the set of all elementals i such that i is less than 4"). How do A and B relate? Well, it seems like any element of A is also an element of B since any number less than 4 is definitely less than 5. Whenever a set shares all its elements with another set, we say that it is a subset of the other set. i.e. A is a subset of B in this example. You can think of A as living within B, like this image below: