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lec0123 - Set Theory A set is a collection of objects or...

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Set Theory A set is a collection of objects, or elements . Typically, the type of all the elements in a set is the same. (For example, all the elements in a set could be integers.) However, it is possible to have different types of elements in a set. (An analogy for this is that usually a bookbag contains just books. But sometimes it may contain other elements such as pencils and folders as well.) We have two standard methods of denoting the elements in a set: 1) Explicitly list the elements inside of a set of curly braces({}), as follows: {1, 2, 3, 4} 2) Give a description of the elements in a set inside of a set of curly braces as follows: { 2x | x N }. In order to understand the second method, we must define the various symbols that are used in this notation. Here is a list of the symbols we will be using: | - translates to “such that”. - “is an element of” - “is a subset of” (Note: in the book they define this symbol as “is a proper subset of”, and use to mean “is a subset of”.) Now we have to define what a subset is. A subset is also a set. So, if we have sets A and B, A B if for all x A, x B. In layman’s terms, a set A is a subset of a set B, if all the elements in the set A also lie in the set B.
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We still have to define what { 2x | x N } really means. Here it is in English: “The set of all numbers of the form 2x such that x is an element of the natural numbers.” (Note: The set N denotes the natural numbers, or the non-negative integers, according to the book.) So, the set above could also be listed as {0, 2, 4, 6, ...} Now that we have gotten that out of the way, let’s talk about the empty set( ). The empty set is a set with no elements in it.
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