Sets_and_Counting - A Note About Sets And Counting There...

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A Note About Sets And Counting There are many files and videos. You need to read and view all of them! Practically every homework problem assigned to you is developed at least in part somewhere in these readings. Some of your homework problems require you to extend or combine concepts beyond the reading. There hasn’t been a new or unique word written about sets in two hundred years. Use your textbook to expand your example base. You will find similar examples in any standard textbook about sets and counting!
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An Introduction to Sets The concept of a set is intuitive. We view a set as a collection of objects. Nothing more, nothing less. However, when we begin to work with sets in a practical way, we hope that we can find a relationship between those objects. It is the relationship that we try to verbalize or otherwise quantify. Let’s look at some examples. Here’s a picture of a room. To begin the process, we could describe a set this way: “Let R be the set of all the objects pictured in this room .” The words objects, items and elements are used interchangeably. Notice that since we have the picture , this is a well-defined set . Without this picture, the description is very vague. Also, if I say that R is all the objects in this room , I would be seriously incorrect. There are many other items in this room not shown in the picture. The word “pictured” is critical to this definition. A well-defined set must convey the same sense of the collection to all users. To help us do that, we list an element of a set exactly one time in its description. We’ll see why when we get around to counting processes. However, the order of the listing doesn’t matter . Now let’s talk about membership in a set. We regularly talk about the elements of a set . Since we used R as the name for the set of objects pictured , it is natural to let r represent an item pictured in the room. So some of our choices for r could be a television , a Japanese doll , a model ship , and a small spoiled dog named Princess . (Princess is a Pomeranian, but she does not know she is a dog! She invited herself into this picture.) If we compile these elements into a set by themselves, we have a subset of the original set R . Let’s call it A . Set Notation It’s time to introduce some set notation . The following phrases and symbols are regularly used in talking about sets: The phrase “ the set consisting of” is represented by braces : { } Example: R = { all objects pictured in the room } Example: A = { television, Japanese doll, model ship, small spoiled dog } The phrase “ is an element of ," or more simply “ is in ," is represented by the “E”-like symbol 0 . Example: Princess 0 R means “Princess is one the objects pictured in the room.” Example: television 0 R means “A television is one the objects pictured in the room.” The phrase “ is a subset of ” is represented by the “C”-like symbols d or f . (Think “is C ontained in”) The d symbol means that there is at least one element of the larger set not in the subset. This is also called a proper subset.
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This note was uploaded on 06/16/2011 for the course MAT 211 taught by Professor Seal during the Summer '08 term at ASU.

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Sets_and_Counting - A Note About Sets And Counting There...

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