Math

Here we dig into ordinal and cardinal arithmetic a little bit. We also establish the famous result (originally due to Cantor) that |S| < |P(S)|, i.e. that the power set of any set is larger than the set.

In this entry, we establish more results about ordinals, including proving that every well-ordered set has the order type of an ordinal. We introduce transfinite induction, Cardinality, and use the Axiom of Choice to prove that every set can be well-ordered, among other results.

In this article we introduce and discuss all of the axioms of ZFC. We introduce Cartesian products, linear and well-orders, and ordinals.

In the previous entry, we introduced Rings and Fields. In this entry we continue our investigation. We will investigate structures that can be built on top of them. This entry will mostly involve grinding through fundamental proofs that are necessary for more interesting results which will come later.

In this entry we introduce rings and fields. We establish some of their basic properties.

In this article we combine some Linear Algebra with a bit of geometry to make some real-world calculations about finding the nearest line to two or more points on a map. We’ve also implemented a tool that uses those calculations to fit a great circle to two or more points on the globe. Check it out!

In this entry we prove that A_n is simple when n >= 5. We then use this result to prove that there is no generalization of “even” and “odd” permutation to three or more classes – i.e. there is no class of homomorphisms into Z_n, where n > 2.

This entry starts out by building to Zassenhaus’ Lemma, and finishes by proving the Jordan-Holder theorem about subnormal sequences for finite groups. Enjoy!

In this article we mostly talk about partial orders, and lattices, which are a special type of partial order. We tie it in with our discussions of subgroups before proving two distribution laws for lattices.

We’re continuing our proofs of Sylow’s Theorems, and also continuing our efforts to count orbits, and determine the size of orbits created by the permutation groups…