Set Theory
Throughout other posts, I have mentioned “sets.“ From infinity comparing the sizes of sets to manifolds finding similarity between them. But what exactly are they and how can people devote full careers to studying them? Ins et theory, there are a couple of fundementals that lay the grounds for the rest of the sets. These are known as axioms.
The first most common axiom is that two sets with the same elements are equal. This is relevant when finding similarities. A set (A), written as {-2, -1, 0, 1, 2} and the set (B), {-2, -2, -1, 0, 1, 1, 2}, both contain the same “elements.“ Even though set (B) has repeating elements, these two sets are still equal. Order also doesn’t matter when comparing elements in sets.
The next most common axiom is that there is a set with no elements called the “Null“ set. This null set is most important when comparing other sets. The following examples use the Sets X and Y.
X U ∅ = X
X ∩ ∅ = ∅
X ∩ Y = ∅ if X and Y are disjoint
There are many other properties of the null set that aren’t immediately useful, but one important note is that any non-empty set has at least two subsets, one of which being the null set.
So how are sets applied? One basic way to look at sets is the idea of a function. If two sets have a one to one matching, they are a function. If there exists one element in either set that has two matching elements in another set, then that is not a function. This is exactly like graphing functions and looking at the x and y coordinates. If a graph has a unique y-value for every x-value, it is a function. If there exists an x-value with more than one y-value, the graph is no longer a function. This same theory is also applied to manifolds and other shapes and is not just limited to two-dimensional graphs.
If you have read any other post, the two axioms above should seem familiar. The top one directly relates to the concept of infinities along with the donut to coffee cup problem. The second one also relates to topology and the comparing of two different manifolds. These basic set laws are so fundamental to math that they appear from graphing to high-level physics. Without these basic axioms, along with a couple of others, mathematics would have no foundation. But if there are already discovered axioms, how can there continue to be set theorists?
The simple answer is that any mathematical anything can be “sorted“ or contained in sets. Set theorists then find similarities between different sets along with discovering and proving more axioms. Set theory is also seen as the common ground connecting different mathematical fields. Even if you do not devote your entire study to just sets, a basic understanding is still required for any other math field.