Set Inclusion NotationDate: 9/12/96 at 19:6:56 From: Anonymous Subject: Set Inclusion Notation Dear Doctor Math, When using the defining property version of set notation: {x|x<2}, why is the first "x" necessary? Why can't we just say {x<2} to read as "the set of all x's less than 2", instead of "the set of all x's such that x is less than 2"? Date: 9/12/96 at 19:12:54 From: Doctor Jodi Subject: Re: Set Inclusion Notation Hi there! You're question is a very good one. I think the problem is that a set can look like this: {5, elephant, 7/9, Aegean Sea} So your set {x<2} could mean that the expression "x<2" is part of the set. Though it is tedious to have to write {x|x<2} out, it's important to make the distinction clear. Let us know if you have other questions. -Doctor Jodi, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 9/12/96 at 21:49:10 From: Louis Feicht Subject: Re: Set Inclusion Notation Dear Dr. Math, In the set {5, elephant, 7/9, Aegean Sea}, an elephant is a member of the set, not the "e", "l" etc. It is what the word represents, just as x<2 doesn't mean the symbols "x", "<". I'm not sure I understand yet. Lou Date: 9/12/96 at 22:49:12 From: Doctor Pete Subject: Re: Set Inclusion Notation Here's another way of looking at it. Say you have the set {5+7,12}. Although 5+7 = 12, clearly the purpose in writing the above expression instead of {12,12} or {12} is to emphasize the addition operation on the two numbers 5 and 7. By the same token, we can also say {1+11,4+8,5+7} and this is quite different than simply saying, {12,12,12}. Now, most of the time we wouldn't be interested in the different ways of writing 12, but this was just an example of how equivalence doesn't preserve information; that is, in the process of replacing 5+7 with 12 you lost the original "5" and "7" which is different from "4" and "8". Here's something that more closely illustrates your original question: What is the difference between {x and y in Z+ and x+y = 12} and {x | x and y in Z+ and x+y = 12} ? The first reads, "The set containing the criterion 'x and y are positive integers whose sum is 12'" and the second reads "The set of all x such that x and y are positive integers whose sum is 12." There is a difference. You could interpret the first set in two ways: You can choose to evaluate the criterion, or leave it alone. But if you evaluate it, what quantity is it that is required from this condition? x? y? x+y? x^2+y^2? Because this is ambiguous, we can only interpret the first set as a single-element set which contains an element which is a condition. On the other hand, the second set is unambiguous, and is equivalent to {1,2,3,4,5,6,7,8,9,10,11}. Does this make sense? Let me apply the same principle to your original statement, {x < 2} and {x | x < 2} . The first says "The set containing the criterion 'x is less than 2'" and the second says, "The set of all x such that x is less than 2." What's the difference? Well, in the first set, *you cannot assume that what you are asking for is x!* How do you know that what you want is x and not x+2? I could say, {x+2 | x < 2}, which under the reals is equivalent to {x | x < 4}. But simply saying {x < 2} tells us nothing of how to apply this condition. It is not right to assume that all we want is x, because if that were the case we should have written {x | x < 2}. Does this make sense? The point is that trying to apply the condition in {x < 2} is not possible unless you know how, which is precisely what {x | x < 2} tells you. Because this ambiguity arises, you must treat the first set as a set containing one element which is the condition "x < 2". Notice I never treated "x < 2" as something composed of pieces, like "x", "<", etc. It is a condition, a restriction as to what values x may take. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 9/13/96 at 13:21:15 From: Louis Feicht Subject: Re: Set Inclusion Notation Thanks, that helped a lot! I'm still going to have to think about some more examples before I can explain it to eighth graders Lou |
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