Examples of sets include the set of integers, the set of lines in a plane, and the set of all dog breeds. The set of all dog breeds is an example that I threw in there to demonstrate that sets can be composed of anything at all. We will never deal with it again.*
The sets that we will be talking about are unordered even if they are usually thought of that way. For instance, there is a well known ordering that comes with the integers - 80 is greater than 4 - but here we just assume that the set of integers is just a big bag of numbers where nothing is ranked. Also, sets will not contain repetitions of objects - everything in a set is going to be distinct. More on this last point later.
*Except for right now:
Problem 1: Is the set of all cute pug puppies equal to the set of all things that deserved to be photographed?
Note: The original first problem discussed in the comments follows in italics, but was changed due to protest and also the fact that it uses polynomials, which most people hate, when it doesn't have to.
Problem 1: Is the set of the roots of the polynomial x2 - 1 equal to the set {1, -1}, that is, the set that contains 1 and -1? (This probably seems like a trivial problem, but don't overthink it, just try to explain why.)
I'm a little thrown by the precise meaning of words here...not blaming you, but I need some clarification.
ReplyDeleteAt first I thought, no, the sets are not equal, because there are an infinite number of sets that *contain* the roots of x^2 - 1. For example, {Z}, or {1, 0, -1} contain the roots to that polynomial, but they are not equal to the set {1,-1}.
But if it's understood that when you say "the set that contains the roots of the polynomial x^2 - 1" you mean "the set that only contains...", then yes, the sets are equal. They are equal because...well, I guess because a set is wholly defined by its elements? It ain't anything besides that; there's nothing to be said about the box itself, so to speak. And the set that consists solely of the roots of that polynomial contains exactly the same elements as the set {1,-1}.
Or am I missing the point entirely?
Also, I saw the first part of this post in Google reader initially and I can't express to you how disappointed I was that dogs were not a part of the problem.
ReplyDeleteYes, you're right. I should have said "the set of the roots..." rather than throwing the word contain in there. I will try to be more careful as we move forward.
ReplyDeleteAnd, yes, your answer is dead on. I'm sorry that the first problem was kind of shallow. I guess that I was trying to move you towards answering the question, "what does it mean for two sets to be equal?" And you said it - all of their elements are the same. In general, we can say that two sets, A and B, are equal if every element of A is in B and every element in B is in A. If we only know that one of these is true (every element of A is in B but perhaps not the other way around), then we have a subset (A is a subset of B). So another way of saying that two sets are equal is to say that they are subsets of each other. So from this loose definition, any set is a subset of itself. At the other end, a set that is completely empty of elements is also a subset (of any set), and it is called the null set.
I'm sorry if this seems like child's play or if it's repeating too much of what you already know. I probably could have just blurted out all of these initial definitions and everything would have been cool, but I wanted to launch into a problem right away and see what shook out. Sorry about the dogs. I'm also disappointed.
also, i just edited the problem to take out the contains bit to any potential future problem solvers confused about the discussion!
ReplyDelete_________________________________
ReplyDelete[ More puppies fewer polynomials ]
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Fine, ethan!
ReplyDeleteIs that meant to be a flag, Ethan?
ReplyDeleteI had trouble with this at first because I was tripped up by vocabulary. I guess I'm just writing this to let you know I'm here.
which vocabulary? pug puppies?
ReplyDeletehow can i help?
ReplyDelete"Roots" and "polynomials," I guess, specifically. It's been since 2005 that I've had a math class, and it took me a moment to remember the rules I was taught about those. Just a moment, though. And then I could see how that wasn't the point very much.
ReplyDeleteI finally got around to reading that Lockhart essay you showed me (was putting it off because I'd already read an opinion piece that mentioned and summarized it). It's not very often that I feel inspired and depressed at the same time. What a mess.
I'm taking a College Algebra course that starts in a few days. Did I mention that already? I'm hoping it gets my logic up to snuff for some electrical engineering classes in the fall; I haven't thought this way in a long time. I spoke to the department head about getting advised about this plan, and before we arranged a time, he asked me, "How is your math?" I didn't know how to answer. Do you think he meant how am I at painting by numbers, or my capacity for imagining up patterns and such? In either case, I think it made him happy that I mentioned Hendrix.
Oh, right - the original problem 1 was complete bullshit, and I'm mad at myself for putting it up there. But I have to leave because otherwise the comments wouldn't make much sense. Anyway, do the new problem 1 which might seem like a joke but it's not really. With the first problem I basically just wanted to address the question, "What makes two sets equal?" So throwing roots of polynomials in there was stupid times.
ReplyDeleteIf you did want to do the roots problem though, then know that roots are just the x value that makes the expression zero (the expression here being the polynomial, x^2 - 1). You might find the new problem 1 to be easier when I tell you that I do believe that all cute pug puppies should be photographed but other things are deserving of being photographed as well.
Thanks for reading the Lockhart essay. It made me feel the same way. Hopefully someday I will be in a position to try and do something about all of that even in a completely insignificant way.
You're taking college algebra at UALR? Who is your professor? I hope that you find it enjoyable. I'm sure that you'll outclass everyone else in there. And, yeah, for electrical engineering, he probably meant paint by the numbers, but, of course, being able to use creativity within a logical framework would serve you well there probably.