Class Review: Okay, get ready for some more sets on steroids

First, I’ll start out by commenting on something I didn’t realize in the last class: A statement can be both conditional and an open statement. Whew. Feels good to get that off my chest..moving on. x

So far, we’ve been talking a lot about sets. Would you even be surprised that you can do even more with them than we already have? I mean, I was when I first found out, but you can pretend that you knew ahead of time or something, whatever.

So sets. You can “union” them (yes, it’s a verb now), you can intersect them, you can subtract them. GUESS WHAT THERE’S MORE.

It’s this concept called Cartesian Product and it will blow your mind. So you have two sets, right? Say A and B for instance. And you want to multiply them, or at least multiply them in a way sets can be multiplied. You set it up like:

A x B

Simple enough for starters. But now you have to solve it. When you take the Cartesian Product of two sets, you combine the 1st element of A with the first element of B, and then the 1st element of A with the 2nd element of B and so on, then the 2nd element of A with the 1st element of B, and so on. You’re basically creating little sets of combined elements of both sets, but it’s important that the elements from the first set listed are listed first in each new set made, and the elements from the second set listed second, and so on. Make sense? An example will help.

{1,2,3} x {$,☼} = {{1,$},{1,☼},{2,$},{2,☼},{3,$},{3,☼}}

See what I mean now? It’s important to note that the Cartesian Product consists of little, 2-element sets, not just combined number and symbols.

Okay, mind blowing example coming up, hold on to your seats ladies and gentlemen.

ℝ x ℝ

Alright, no big deal. You just start it out nice and slow. The first set is {1,1}, {1,2}, {1,3}…what this keeps going, there’s a lot of real numbers. Wait there’s a ton of real numbers, there’s an infinite amount of real numbers what do you do what do you do?

First of all, calm down. Second of all, just write it out like this:

{ (x,y): x,y ∈ ℝ}

See you’re fine. BUT. Guess what? If you graph his Cartesian Product, you get….this!

14by14 axes

The x/y plane that we use ALL THE TIME in math? IT’S THE CARTESIAN PRODUCT OF TWO SETS OF ALL REAL NUMBERS.

Now let’s get even freakier. You don’t have to stop at just crossing two sets. You can cross as many as you want. Like this!

ℝ x ℝ x ℝ x ℝ x ℝ x ℝ x ℝ x ℝ x ℝ x ℝ = ?

This Cartesian Product looks like the x/y model, but with 11 axes. FREAKY. Also, to save yourself the writing (mathematicians are all about that), you can write that product as:

ℝ x ℝ x ℝ x ℝ x ℝ x ℝ x ℝ x ℝ x ℝ x ℝ = ℝ11

Okay, now seriously prepare yourself, cause now we’re going to talk about power sets.

Power sets are basically a set of subsets that are included in a given set. An example would be better to show this instead of just using the word set 3 times..

P({1,2}) = { {  }, {1}, {2}, {1,2} }

See what I did there? The set of {1,2} is made up of the set 1, the set 2, the set itself, and of course, the null set. In fact, the null set is always a subset of any set.

Complements are something else too. Complement sets are a set of elements that are NOT in a given set. To express this, we need a universe that we can pull from. For example,

{1,2,3}’ (universe = {1,2,3,4,5} = {4,5}

{  }’ = universe

Universe’ = {   }

And this is the point in class where everything went crazy. Did you know you can take the power set of a power set? You can take a union of two power sets? You can take an infinite amount of cartesian products and then union or intersect them? What am I even talking about..it’s Saturday. We’ll take about it on Monday.

Happy Saturday ya’ll!

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