But it's not bland. And it's certainly not pointless! In fact, the set of real numbers, R, contains such an infinite amount of points that it makes the infinite set of integers feel insecure about their petty and miniscule size. (Yes, a math joke or two just went on here. Kill me.)
As for pragmatic: abstract algebra most certainly is...sometimes. But, dude, "sometimes" is better than "never," so back off already. Jeez! (As we math fans like to say, it's always "interesting in it's own right," but unfortunately you have to first think it's interesting to come to that conclusion...and that means putting in the time to understand what the hell is going on.)
Abstract algebra has applications in almost any kind of mathematically inclined science imaginable: particle physics, quantum mechanics, computer science, engineering, robotics, electromagnetism, chemistry, and (drum roll, please) even biology.
The last one is interesting and, by the way, is the reason I set out to write this blog series. If you are asking yourself, how the hell can abstract algebra apply to biology, you're not necessarily stupid or clueless (although, who knows, you might be). The connection seems awkward at first, a bit outrageous! But it's possible. To get there though, we must trek through the myriad landscapes of abstract algebra, which isn't a walk in the park.
Biology is the study of some highly complex systems, so right away you can imagine that it may not be easy to uncover mathematical structures, or once uncovered it might not be straightforward what, if anything, this adds to the study of biology. For example, there exists a structure based on non-Mendelian genetics called "evolution algebras." These algebras are "nonassociative." In plain-speak this means they're nasty little suckers. But what, if anything, have evolution algebras said about evolution? Anything new? Anything much at all?
To be honest, I'm not yet sure. I am still looking into it myself. Over the next few months, I'll try to get a hold on the current state of affairs. Ultimately, maybe I'd like to push those affairs forward. But we'll see.
Just to motivate any biologists out there a little more, there exist algebras with cool names such as "gametic algebras," "zygotic algebras," and "genetic algebras." (Check out the intro this 1939 paper to get an idea here.)
Anyway! Before we get there, we have to continue building up our vocabulary and know-how. By going over some cheesy scenarios in the past two weeks, I've aimed to initialize that first stab at familiarizing some basics.
As a recap, we addressed equivalence relations, equivalence classes, a slight intro to group theory, some talk about the integers, and quite some discourse on modular arithmetic. I've mentioned, without covering, words like quotient groups and rings.
Somehow, over the coming weeks, I really want to try to get a little more formal and rigorous, but without losing the casual tone.
That means I have to come up with a plan.
On the one hand, I don't want any of this to be like a text book. But on the other, I don't want to be so spotty that incoherence inevitably ensues.
So maybe a plan is indeed in order: for starters, we need to attempt to establish the basic terminology of set theory. That's the first step towards really getting the most out of the abstract algebra experience.
A "set" is the most basic object in mathematics. So basic that there's no proof of a set's existence (besides our intuition) or even a rigorous definition: a "set" is a "collection" of objects. See? It's pretty much circularly defined. We can't do much about that. But accepting this as a starting point, we can go pretty far. (Or, in more rigorous terms: we can go pretty DAMN far.)
So say we have a set S. If it's a small, finite set, we often choose write out its elements: e.g. S = {a, b, c}.
For a bigger set, we usually attempt to write down conditions which all elements of S must satisfy. For example, if S is the set of integers greater than 10 and less than -8, we can write S = {x∈Z: x>10 or x<-8}, which reads, "S is the set of all integers x such that x is greater than 10 or x is less than -8." (Remember: we usually reserve capital Z to denote the integers.)
To be clear, the symbol ∈ means "is a member of" or "is an element of." So x∈Z means "x is a member of the integers" or more simply that "x is an integer." The colon within the set brackets reads "such that."
A set doesn't have to be finite, for example let S be the following set: {x∈R: 1<x<2}. This is an "open" set containing all the real numbers (capital R is often reserved to denote the reals) between the numbers 1 and 2, without actually containing 1 or 2. This is an infinite amount of set elements. No matter how close we get to the number 1, we can squeeze another real number in without touching 1.
A subset T of a set S is simply another set that contains only elements of S. We write: T ⊆ S. We might say, "T is contained in S," but how we say it out loud usually is something more loose, like "T sits in S." This symbol kind of looks like the "less than or equal to" symbol, which is a decent comparison: as far as sets go, it literally means T is either all of S or just a part of S.
An example of a subset is the set of integers embedded in the reals: Z ⊆ R. We can actually write this relationship using a stronger symbol: Z ⊂ R. This reads, "Z is a proper subset of R." (Or, again, loosely we might just say "Z sits in R," mentally noting that it does so "properly.") This means what you might think it'd mean: the set Z is strictly contained in R, i.e. there is no need to suggest the possibility of equality of these two sets: the integers are definitely not the entire set of reals.
It's important to remember that a set needn't be a set of numbers. It can be a set of any kind of objects---an example from phase 2 was "the set of all annoying people in your life." Obviously the "set of all annoying yet attractive chicks you know" is a proper subset of such a set.
To be thorough, be warned that these set relations are sometimes written backwards like so: let S be a set and T a subset of S, then T ⊂ S can also be written S ⊃ T, which you might say aloud as, "S contains T."
Given any set S, there exist two "trivial subsets." We call them trivial because they tell us nothing very important or unique about the set in question. One trivial subset of S is S itself: S ⊆ S. This tells us nothing new. The other trivial subset of any set is the "null set," denoted by ∅. This is also known as the "empty set." It contains nothing. Written using the bracket notation: ∅ = {}.
Remember the requirement of being a subset: "T is a subset of a set S" means that all the elements of T are in S. Does the empty set meet this requirement? Well, we can confidently say that it contains no element outside of S. (Obviously, since ∅ has no elements.) So we say that the empty set vacuously satisfies the requirement of being a subset of any set S.
There are operations on sets too. For example, say you have two sets and you just want to combine them: we call this a "union of sets."
Here's the gist: given a set A = {2,4,5} and a set B = {E,F,G}, the union of A and B, written A∪B (more loosely: "A union B"), is the set {2,4,5,E,F,G}. As another example, say you have the set {1,2,3} and the set {2,3,4}, then the union of these two sets is just {1,2,3,4}. You might be tempted to write this new set as {1,2,3,2,3,4}, but given a little thought, this says nothing more about what elements the new set contains than does {1,2,3,4}.
Suppose you want to find out what elements are common between two sets: we call this the "intersection of two sets." So, if you have S = {z∈Z: z>10} and R = {z∈Z: z <100}, the intersection of S and R (or more loosely: "S intersect R"), written S∩R, is the set {z∈Z: z>10, z<100}, which reads, "the set of all integers that are greater than 10 AND less than 100." (So this is the set the contains the integers 11 through 99.)
There is a bit more to pick up about set theory, so try taking a look at Schaum's Outline of Set Theory on google books and also, the first chapter or two of Schaum's Outline of Abstract Algebra. If you want to purchase an awesome, inspiring, very easy-to-read intro to abstract algebra, I urge you to purchase the book by Charles C. Pinter.
I think we currently have the machinery to construct the natural numbers out of nothing. (The natural numbers are usually denoted N. This is the set of all positive integers, i.e. the "counting numbers.")
Start with the null set, ∅. The next step is create another set like so, {∅}. Then we create another set which contains the previous two as elements: {∅, {∅}}. Again, create another set---this time with the previous three sets as elements: {∅, {∅}, {∅, {∅}}}. Continue to construct sets in this fashion, always including all the previous sets as elements of the new set.
Bam! We've constructed the natural numbers from nothing (more specifically, we've constructed N∪{0}, which might read aloud as, "N unioned with the singlet set of zero" or less anal as "N union zero"). They look a little different than the symbols we're used to, but we can create a one-to-one correspondence (a "bijection") between sets to show the construction explicitly:
| ∅ | → | 0 |
| {∅} | → | 1 |
| {∅,{∅}} | → | 2 |
| {∅,{∅},{∅,{∅}}} | → | 3 |
| : |
Is this a trick, or is does it dig deep into the fabric of the universe? Just keep thinking about it as you fall asleep. If anything, it's something to ponder.
-Kevin
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