functions

We’re Still on Grade School Stuff Remember back in grade school when the teachers taught you about functions? They were always represented as $f(x)$ (which apparently is according to Euler? Idk some YouTube video said its his notation) and were discussed as some sort of “input/output machine”.

Some of Math’s Coolest Terms We have the general idea of functions, but as with anything in math, we can define some useful properties to have. Injectivity We have as a propery of functions that every input must have exactly one output, but what if that output is always unique to the given input?

Data Structures We have already talked about how functions in computer science, and how they are also just as functions are in mathematics, but there are many more applications of functions than just this.

The Hardest Math Theorem ☠️ The pigeonhole principle is probably the theorem in math that scares me more than anything else in the world, because it is so powerful and fundemental, but actually applying it can be some of the most challenging problems a math student can face.

What does it mean to count? When I ask you “How many apples are there” depending on however many apples you are counting, you’ll mentally provide some number. $1,2,3,4,\ldots$ etc. Counting is counting, you’ve done it since you were a child.

The Rationals and the Reals On the previous page we showed (kinda) how the rationals $\mathbb{Q}$ was a countable set the same size as the naturals $\mathbb{N}$. Right now we will take some time to how $\mathbb{Q}$ is quite intertwined with the set of real numbers $\mathbb{R}$.

Relating Things Together Through the use of functions, we can define a lot of the things that we use in general, such as operations like $+$ and $\cdot$. As of right now though, we don’t really have a good way of comparing things together.

Always More Terminology 📖 Now that we have relations, we can define some properties about them and see how our examples from the previous page compare. In all of our definitions, we will use $*$ to represent some generic placeholder symbol, with $R(*)\subseteq A\times A$ being our relation set.