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Set Theory
Putting things into groups? How hard could that be?
Last updated on Aug 15, 2023
Size of Infinity
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.
Last updated on Dec 12, 2023
Larger Size of Infinity
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 $\mathbb{R}$.
Last updated on Dec 12, 2023
Relations
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.
Last updated on Jun 30, 2023
Properties of Relations
Always More Terminology ðŸ“– 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.
Last updated on Dec 12, 2023
Equivalence Classes
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.
Last updated on Dec 12, 2023