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SetTheoryL1.L2.Version2Spring.2011

SetTheoryL1.L2.Version2Spring.2011 - 1 Set Theory The first...

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1. Set Theory The first mathematical topic that we have to look at in order to understand how search engines work is the topic of sets . We need this because the first step that any search engine does in trying to answer your question, is to identify a collection of web pages that in some way seem to match your question. That collection is a set . The way that the engine finds that set involves certain basic mathematical operations that can be performed on sets. So here we go. Definition: A set is a collection of objects. 2 Examples of collections of objects: My cat, my dog, and my elephant. All cats. The animal types {cat, dog, cow, tiger} All students in ITI 111 The students {John, Maria, Pedro, Paul}. Students whose names are John, Maria, Pedro and Paul. 3 There are two important ways to specify a set, by a list, and by a rule. The first way to specify a set is by listing all of the objects that are in it. This is done, on a page, by using curly braces to represent the beginning and end of the set and listing the things that are in the set, or their names, usually separated by commas. So we might have a set that contains the letters a,b,c,d,e,f,w. We could give that set a name (C) and write in equations something like C={a,b,c,d,e,f,w}.
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4 The second way of specifying a set: giving a rule that tells what belongs to the set. The way of writing that rule also uses curly braces, but it uses the colon punctuation mark to stand for the words “such that”. So if we write something like: { : } B letters theletterisavowel , then we have given an expression that means “the set of letters that are vowels”. You would read the above as “a set of letters such that the letter is a vowel.” HOW ELSE CAN WE WRITE THIS SET? B= {a, e, i, o, u} It‟s important to not e that the order in which the elements of a set are listed does not matter. The set containing the letters a and t is exactly the same set as the set containing the letters t and a. So { , } { , } a t t a Every element in the set is unique. So {a, a, t, t, t} = {a, t}. Membership notations: x in X. In addition quite sensibly, we say that two sets are the same if and only if their unique elements are exactly the same (regardless of order) . This is the definition of “equality of sets”. 5 Notice that I have snuck in a new mathematical word, element” here without even announcing it. If you ever studied chemistry you learned that the word “element” means something like Hydrogen or Helium or Carbon. In mathematical language we use the word “element” as another way of talking about the things that belong to a set. It‟s not the only word we will use. Sometimes we will use the word “member” and we will say that for example “ a is a member of the set { , } a t ”. There are other w ays to say it also. “a belong to the set…” or “the set contains a…”.
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