Infinite Proper Subset of an Infinite SetDate: 09/22/97 at 02:43:20 From: Sam Cotten Subject: Infinite proper subset of an infinite set My wife and I are in a Math survey class for Elementary school teachers, (she teaches K, I teach 2nd grade.) A question on a recent test was: Given set A= {1,2,3,...} and set B= {10,20,30,...}, is B a proper subset of A? The book we are using, _Fundamentals of Mathmatics_, by William M. Setek, states that "A proper subset contains at least one less element than the parent set." My reasoning was that since the two sets are both "countable" infinite sets, and can be placed in a one-to-one correspondence, they are therefore equal and B does not have one less element; therefore B is not a proper subset of A. I was marked wrong on this question, and the professor simply stated that "B is contained in A, and is therefore a proper subset of A." I guess my question has to do with the actual definition here: is the book wrong, or at least worded improperly, or am I wrong in thinking that the fact that two sets are the same size, i.e., they have the same number of elements, precludes one from being a proper subset of the other? It seems to me that it is merely a subset, not a proper subset (?). Should the book not say something like "an element not contained in..." rather than "...one less element"? Date: 09/22/97 at 08:18:45 From: Doctor Jerry Subject: Re: Infinite proper subset of an infinite set Hi Sam, Two sets can have the same number of elements and not be equal. The set P={a,b,c,...,z} of 26 letters has the number of elements as the set Q={1,2,3,...,26}, but the two sets have no elements in common. The sets A= {1,2,3,...} and B= {10,20,30,...} have the same number of elements (there is a one-to-one correspondence) and B is a (proper) subset of A. The usual definition of subset is: A set A is a subset of a set B if each element of A is an element of B. A set A is a proper subset of a set B if A is a subset of B and is not equal to B. I think the phrase "A proper subset contains at least one less element than the parent set" is not widely used. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 09/26/97 at 12:57:17 From: Doctor Ken Subject: Re: Infinite proper subset of an infinite set Hi - Let me add a little bit to this. Strictly speaking, if your textbook only talks about finite sets, its definition is fine. But if it talks about infinite sets (as your professor does) then it is incorrect, for exactly the reason you stated: to properly apply the criterion to two sets, you have to count up the number of elements in each set and compare the two numbers you get. This technique fails in the case you talked about. A more suitable definition of proper subset would be something like this: A is a subset of B <=> every element of A is an element of B A is a proper subset of B <=> A is a subset of B AND there is some element of B that is not an element of A or equivalently (as Dr. Jerry wrote) A is a proper subset of B <=> A is a subset of B AND A does not equal B Under this more correct criterion, we can see fairly easily that your set {10, 20, ...} is a proper subset of {1, 2, ...}. So it's quite common for a set to be a proper subset of another set of the same size. Your professor should recognize that your book's definition is not very good for infinite sets. At the very least it's ambiguous about what's supposed to happen with infinite sets. -Doctor Ken, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 09/26/97 at 16:30:54 From: SAM COTTEN Subject: Re: Infinite proper subset of an infinite set Thanks very much for writing again. I talked to my professor about this and was able to convey, though not as well as you have, what I thought the problem in the book was. He understood my confusion, but isn't going to give me credit for the question I missed as a result... Interestingly, the book does not specifically address infinite sets in this context, nor did he (the professor). It was just on the test, and we were supposed to figure it out with the knowledge gained. I think quite a few students shared my confusion. To him, and other math-oriented people I have discussed this with, the answer is obvious; so obvious that they didn't think they needed to explain it. Perhaps the author of the book felt the same way. Some of the interesting fallout of this situation is that: a) I had to find out about this on my own - something we're always exhorting our students to do - and b) now I understand a lot more about this fascinating subject than I did before. So it's been well worth it from that viewpoint. I have not been in a pure math class for over 25 years, and a lot has changed in the way math is taught. "New Math," as it was called then, was introduced about halfway through my high school years, and I'm afraid many of the teachers of that time were unable to really comprehend it themselves, much less teach it effectively. I studied set theory only briefly in one course (Algebra II?) in high school; now, we use Venn diagrams in first grade onwards for everything from math to language arts (they're great for comparing stories, etc.) I only regret not getting this kind of foundation early on myself, as it is becoming very clear how these concepts form the basis for all the math I did take - and I took a lot of it! I'm only now beginning to understand things I only did by rote then... So thanks again, and keep up the good work, I'm very impressed with your program! Sam Cotten |
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