MGF3301

MGF3301 - Introduction to Advanced Mathematics Course notes...

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Introduction to Advanced Mathematics Course notes Paolo Aluffi Florida State University
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1. AUGUST 30TH: BASIC NOTATION, QUANTIFIERS 1 1. August 30th: Basic notation, quantifiers 1.1. Sets: basic notation. In this class, Sets will simply be ‘collections of elements’. Please realize that this is not a precise definition; for example, we are not saying what an ‘element’ should be. Our goal is not to develop set theory, which is a deep and subtle mathematical field; it is simply to become familiar with several standard operations and with a certain language, and practice the skill of using this language properly and unambiguously. What we are dealing with is often called naive set theory, and for our purpose we can rely on the intuitive understanding of what a ‘collection’ is. The ‘elements’ of our sets will be anything whatsoever: they do not need to be numbers, or particular mathematical concepts. Other courses (for example, Introduc- tion to Analysis) deal mostly with sets of real numbers. In this class, we are taking a more noncommittal standpoint. Two sets are equal if and only if they contain the same elements. Here is how one usually specifies a set: S = { ········· (the type of elements we are talking about) | “such that” (some property defining the elements of S ) } For example, we can write: S = { n even integer | 0 n 8 } to mean: S is the set of all even integers between 0 and 8 , inclusive. This sentence is the English equivalent of the ‘formula’ written above. The | in the middle of the notation may also be written as a colon, “:”, or explicitly as “such that”. On the other hand, there is no leeway in the use of parentheses: S = ( n even integer | 0 n 8) or S = [ n even integer | 0 n 8] are simply not used in standard notation, and they will not be deemed acceptable in this class. Doing away with the parentheses, S = n even integer | 0 n 8 is even worse, and hopefully you see why: by the time you have read ‘ S = n ’ you think that S and n are the same thing, while they are not supposed to be. This type of attention to detail may seem excessive, but it is not; we must agree on the basic orthography of mathematics, and adhere to it just as we adhere to the basic orthography of English. An alternative way to denote a set is by listing all its elements, again within { } : S = { 0 , 2 , 4 , 6 , 8 } again denotes the set of even integers between 0 and 8, inclusive. Since two sets are equal precisely when they have the same elements, we can write { n even integer | 0 n 8 } = { 0 , 2 , 4 , 6 , 8 } . This is a true (if mind-boggingly simple) mathematical statement, that is, a theorem. The ‘order’ in which the elements are listed in a set is immaterial: { 0 , 2 , 4 , 6 , 8 } = { 2 , 0 , 8 , 4 , 6 }
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2 since these two sets contain the same elements. This is another reason why we have to be careful with the use of parentheses: (0 , 2 , 4 , 6 , 8) means something, and that something is not the set { 0 , 2 , 4 , 6 , 8 } ; it is the ordered set of even integers between 0 and 8. Thus, (0 , 2 , 4 , 6 , 8) 6 = (2 , 0 , 8 , 4 , 6) : the elements are the same, but the orders differ.
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This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.

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MGF3301 - Introduction to Advanced Mathematics Course notes...

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