My plight is nothing new, and neither is the quiet response from the universe: "Quit your crying, Whiny McWhinestein."
What I can tell you a little about, though, is how to take something infinite--the integers--and turn it into something finite: a Yes-or-No number system that, for lack of a better name (dry humor going on here), is called Z mod 2.
Here's the scenario: you're sitting in the library when, across the room, comes in a voluptuous dame. What do you do? There are so many options--probably one for every integer! You could say nothing. You could jump up and down. You could decide to hold your breath until your face turns blue. Fact is, at each instant there's an infinitude of trivial and petty things you can choose to do--bark like a rabid dog and bite whoever's closest!
We'll represent this infinitude by the integers, Z.
With so many options at hand, it can be hard to pick one. Should you punch your computer screen? Or maybe you'd like to tell the guy sitting next you he breathes too loud? Possibly you might consider tripping the beautiful dame as she scurries past you. (You're feeling particularly violent today for some reason.)
It'd be so much easier to just decide Yes or No in a concrete manner. Screw having so many choices! You say aloud: "I'm replacing my old, beat up Z with a new, shiny Z mod 2." (The hot chick thinks you're talking about sports cars as she walks by and thinks you must be cool---if she only knew you were talking about math.)
Last week, we talked about the integers (Z,+) in terms of being a group. From this group, we created another group, namely Z mod 12, written (Z/[12], +), by defining an equivalence relation on the integers, which partitioned them into 12 equivalence classes.
Each of the equivalence classes technically housed an infinite amount of numbers. For example, we showed that the equivalence class denoted [12] housed every multiple of 12 that exists within the integers, which oddly enough is the same size as the entire set of integers (see: Cardinality or a little about Georg Cantor). Likewise, the equivalence class denoted [5] was shown to house all integers that can be written as 5 added to a multiple of 12, i.e. all integers that have a remainder of 5 after being divided by 12.
It was noted in passing that these equivalence classes were denoted arbitrarily, e.g. [12] could have been denoted as [0], and that any member of the equivalence class could equally well have been chosen to denote the entire class since, for example, [12] + [3] = [12] + [15] = [0] + [15] = [15] = [3]. Work all those out for yourself if your little heart desires. The point is that in the language of Z mod 12, every member of a particular equivalence class is on equal footing--i.e. is equivalent. In Z mod 12, we do not differentiate between 0, 12, 24, 36, ..., or any multiple of 12: they are all in the same equivalence class. Likewise we do not differentiate between 3, 15, 27, ..., or 3 plus any multiple of 12, since they too are all identified as equivalent, which is suggested by the notation [3]. And so what we have done is we have literally made an infinite amount of things into 12 things: all the integers into the equivalence classes [1] through [12] (which, by the way, are often written in other circumstances as [0] through [11]).
By the end of last week's discussion we agreed to stop using the brackets since, once it's understood that we are working in the Z mod 12 number system, they are merely an inconvenience. This is just a choice of convention, as is using the brackets. The particular convention of dropping the brackets is to show that in our new algebraic system, the equivalence classes themselves are treated as single objects: instead of thinking of [2] as an infinite class of integers, we can just think of it as an object in and of itself, and thus relabel it simply as 2. By this relabeling of the elements of Z mod 12, it is more easily seen that imposing an equivalence relation on the integers literally reduced an infinite amount of objects to a mere 12.
12 isn't special. We could have reduced infinity to 10, 73, or 1000. It doesn't matter, the process is the same. (To be thorough, that process is: all integers with the same remainder after being divided by x are considered equivalent.)
These types of objects, since they are derived from the integers, retain some of the properties of the integers--for example the properties of even and odd were retained in Z mod 12, but the properties of "greater than" and "less than" were thrown out.
The retention of even and odd is easy to confirm. You could work out each case, e.g. [2]+[6] = [8], [3]+[12] = [3], etc, if you wanted to. You will come to find that any even plus even results in an even, any odd added to an even results in an odd, etc. A table sums this up succinctly:
| + | Even | Odd |
| Even | Even | Odd |
| Odd | Odd | Even |
Concerning the loss of "greater than" and "less than," ask yourself: in Z mod 12, is 8 less than 10? Remember, these things are equivalence classes: this is the same as asking is "[8] less than [10]," which is the same as asking "is [20] less than [10]." The elements in the number system clearly do not portray this familiar property.
Anyway, given a little thought, you can see that "12-ness" of numbers is retained in Z mod 12.
This is easier to see in Z mod 3, where "3-ness" is preserved. If you know anything about music then you may have heard of triplets--this is 3-ness in action. So in the terminology of a meter like 6/8 or 12/8, Z mod 3 preserves if a number is a "trip," a "pa," or a "let."
| Trip | pa | let | Trip | pa | let | Trip | pa | let |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
You can call these properties tripness, paness, and letness. And you can say things about numbers with these qualities, e.g. all numbers divisible by 3 are "let numbers." This is just like you can say things about numbers that have the qualities even or odd, which are the preserved properties of the integers in Z mod 2, e.g. all numbers divisible by 2 are "even numbers." And so evenness and oddness are the properties related to "2-ness" of the integers. Likewise--tripness, paness, and letness are the properties of numbers related to "3-ness."
(And, given some thought, the equivalence classes of Z mod 12 are the properties related to the "12-ness" of the integers. We could call each equivalence class [1] through [12] something neat like we did for Z mod 3 and Z mod 2, but coming up with 12 names is not something I feel like doing and, seriously, probably not something you feel like reading: it would only serve to take up space, kind of like this sentence just did.)
Anyway! Let's throw out every property the integers have to offer except evenness and oddness, which as noted above gives us Z mod 2. We can construct this system in the same manner as we constructed Z mod 12: take all the integers in one hand, an equivalence relation in the other, and---BAM! Clap them together and out fall some equivalence classes.
This time our equivalence relation is "any two integers are equivalent if, after being divided by 2, they have the same remainder" and our equivalence classes are [0] and [1] or, equivalently "even" and "odd."
We can actually call the elements of this set anything we want though, as long as the structure we are considering is preserved. So, in the spirit of what we set out to do, let's call these equivalence classes "yes" and "no."
It is easy to show that the group structure of the integers is technically preserved in Z mod 2 relative to addition (see the even/odd table above), but if we are to care about the group structure, we must interpret what the following means:
Yes + Yes = Yes
Yes + No = No
No + Yes = No
No + No = Yes
We can be a little cheesy here, no big deal. Let's just say this all means that if you *really* think yes, then go with yes. If you are unsure, go with no. And if you are emphatically thinking no, then go with yes. Of course this line of thinking can get you killed, for example: what if you reeeaaallly didn't want to jump off that cliff? It could also get you fat--we all have our weaknesses, e.g. nachos and cheese!
The group structure isn't really helping our cause here. Fortunately we don't always have to care about the group structure. Groups have many applications and are found in various areas of science, math, and nature, but we aren't bound to them in order to have an interesting algebraic system.
So long group structure!
What we want is some property of Z mod 2 that we can use to construct a decent decision-making system: the property we are looking for comes along with the operation of multiplication in Z mod 2. Just like with the integers, Z mod 2 combined with multiplication does not form a group (why not?), but it does constitute what is known as a semigroup. And, better yet, it gives us a good way of getting off our asses and talking to chicks: here's how!
| * | Yes | No |
| Yes | Yes | Yes |
| No | Yes | No |
You can see this is Z mod 2 by replacing all Yes's with 0's.
We can interpret the operation * as following: "on first thought, but on second thought"
So should you talk to the voluptuous dame? If you really want to, but feel to afraid to actually do it, that equates to "YES on first thought, but on second thought NO," which means you gotta get up, Sweaty Palms.
No fear, just put the pressure on her and say, "I need a laugh. Tell me a joke."
This system is pretty straight forward: if you find yourself thinking "NO on first thought, but on second thought NO," then you shouldn't need the operation table above for it to be obvious that you should go with NO, especially since we can easily reinterpret the binary operation as "...is what my lower half thinks, but my upper half thinks..."
If the lower half is saying the same thing as the upper half, don't waste your time.
Let the good times roll!
-Kevin
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