Universal SetsDate: 08/19/2002 at 12:53:41 From: Tim Melton Subject: Universal Sets - explanation I don't quite understand the concept of universal sets. For instance, if I had to place the President of the United States in three universal sets, what are some possibilities? I need a complex yet simple definition with examples. Date: 08/19/2002 at 15:31:06 From: Doctor Peterson Subject: Re: Universal Sets - explanation Hi, Tim. The universal set in any particular context is the set of all objects under consideration. You might be relating the President to other American men, or to other current heads of state in the world, or to past presidents. Then the universal set would be the set of all American men, or the set of all current world heads of state, or the set of all U.S. presidents. Or it might be the set of current residents in the White House, or the set of children of George Bush, or of men named George, or of human beings throughout history. Or, you might not be thinking of the president as a specific person, but just as the holder of an office, and the universal set might be all federal offices. What do these have in common? The fact that they contain the word "all" (or some equivalent), and that they include the element you were asked about. (If the president were a woman, my first example would not work.) A universal set must be big enough that everything you want to talk about (that is, to include in a set) will be within the universal set; but it need not contain every object anyone could ever want to talk about, or that might be mentioned in passing, such as the White House itself. Here is one brief definition from Eric Weisstein's MathWorld: Universal Set http://mathworld.wolfram.com/UniversalSet.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 08/19/2002 at 15:43:45 From: Tim Melton Subject: Universal Sets - explanation When I describe the universal set with an example you provided me, am I correct in doing as stated below? U={American Men} U={White House Resident} U={Men Named George} I just want to make sure I write it out as I should. Date: 08/19/2002 at 16:08:06 From: Doctor Peterson Subject: Re: Universal Sets - explanation Hi, Tim. It depends on the notation you are expected to use, which can vary from formal U = { x | x is an American man } to informal U = all American men I'm not quite comfortable with your notation, because when we use braces, what is inside them generally represents individual members of the set, rather than a description of the contents of the set. Perhaps you can show me an example of the notation your text is using to define sets, so we can find the right level for your notation. I personally would just say it in words unless I was told otherwise; but I haven't been a student in a long time! Incidentally, the last example is likely to be objected to; it just doesn't sound broad enough to be a universal set. I included it as an extreme example, and there may conceivably be a situation in which it might be considered universal, but you should probably choose some really universal-sounding examples to be on the safe side. On the other hand, you might want to stick that in as a fourth, with an explanation, just to test the boundaries of the terminology as used in your class - and to see how good a sense of humor your instructor has. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 08/19/2002 at 15:46:48 From: Doctor Mike Subject: Re: Universal Sets - explanation Hi Tim, Here are 5 examples of such a universal set. Note that BOTH of the presidents named Bush are in sets 1 through 4, but only the current president Bush is in the final one. See? I hope this helps. 1. The set of all people in the Bush family. 2. The set of all people who have been or are the President of the United States. 3. The set of all living mammals. 4. The set of all living males. 5. The set of all people whose father was a president of the United States. - Doctor Mike, The Math Forum http://mathforum.org/dr.math/ |
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