14 Introduction to Sets

14 Introduction to Sets - Handout #14 April 14, 2010 CS103...

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Handout #14 CS103 April 14, 2010 Robert Plummer Introduction to Sets Sets are the most basic of mathematical structures. It is possible to build all of mathematics on just the concept set membership and a small number of axioms. It is important to know that mathematics has such a foundation, but "axiomatic set theory" is too tedious to develop for our purposes. Our approach will be a more intuitive one, where we assume that we know and understand what objects are, what collections of objects are, and what it means for an object to belong to a collection. This approach is called "naive set theory", and in spite of that name, it provides us with all the formality and rigor we will need. We start with the following definitions: Definitions A set is an unordered collection of distinct objects. The objects in a set are called the elements or members of the set. If a set has exactly n members for some non-negative integer n , then we say that the set is finite . The number of members of a set is called its size , or cardinality . The set with no members is called the empty set and is denoted by the symbol or by empty braces {}. A set that is not finite is said to be infinite . The size of an infinite set is greater than any integer. This fits our intuitive notion of a set. For example, we could consider the set of all students registered for CS103. Each student is distinct from all the others, and no ordering of the students is implied. There are exactly 151 students enrolled at the moment, so the cardinality of the set of CS103 students is 151, and the set is finite. None of the TAs are enrolled, so the set of CS103 students who are also TAs is the empty set. By contrast, the set of all non-negative integers is infinite. We can formally define the members of a set in two ways. The first is simply to enumerate the members by listing them within brackets. For example: CS103 TAs = { Karl, Mridul, Steve, Tyler }
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- 2 - The second way to define a set is to specify some condition for membership. This is referred to as set builder notation . For example: Q = { p/q | p, q are integers and q 0}. defines the set of rational numbers. The slash / means division and the vertical bar | is read "such that." This definition means that the set contains every number p/q that satisfies the condition to the right of the bar. Note that 2/4, 3/6, 4/8, . .. are simply alternate ways of specifying the unique member of the set that we normally write as 1/2. Note also that Q is infinite. As another example, the set of positive odd integers {1, 3, 5, 7, 9, 11, . .. } can be defined as: { n | n = 2k+1 for integer some k 0 } Sets are traditionally given capital letters for names. Some important sets are given special names: R for the real numbers Q for the rational numbers Z for the integers N for the natural numbers {1, 2, 3, . ..} The symbol denotes membership in a set, and denotes non-membership in a set. Thus, 2.51 Q, but 2.51 N. The symbol looks somewhat like the Greek letter epsilon, but it is really a special math symbol.
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14 Introduction to Sets - Handout #14 April 14, 2010 CS103...

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