sets

Putting things into groups? How hard could that be?

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.

Why Duplicate Effort? Suppose we have some equivalence relation $\approx$; note here $\approx$ does not mean “approximately equal”, it just is a generic symbol for any equivalence relation. Since $\approx$ is an equivalence relation, then we know that it is reflexive, symmetric, and transitive.