06. Sets 1

06. Sets 1 - Maggie Johnson Handout #6 CS103B Sets Key...

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Maggie Johnson Handout #6 CS103B Sets Key topics: Introduction and Definitions Set Operations Set Identities Cartesian Product Proofs about Sets Set Applications The first major abstraction that we will consider is a set. Sets are the most basic of mathematical structures. They are so basic, in fact, that they are hard to define without using the word “set” or some synonym of “set”. The set concept allows us to talk about a collection of objects, ignoring order and ignoring repetition. The only relationship between the elements of a set is that they all belong to the same set. We can speak of the set of integers (which is infinite), the set of real numbers (which is infinite in a different way), the set of good pizza places in Palo Alto (very finite), or the set of good pizza places in Libertyville, IL (empty set). Definition: A set is a collection of distinct objects without repetition and without order. The elements of a set are referred to as its members. We can formally define the members of a set in two ways. The first is simply to enumerate the members, as in: P = { Ramonas , Pizza-A-Go-Go }. Note that sets are traditionally given capital letters for names. Some important sets are given special names such as: R for the real numbers Q for the rational numbers Z for the integers N for the natural numbers The symbol denotes membership in a set, and denotes non-membership in a set. Thus, Ramonas P , but Jiffy-Lube P , meaning that Ramona’s is a good place to buy pizza, but Jiffy- Lube is not. A set with no members is denoted by empty braces {}. 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 Z }.
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defines the set of rational numbers. The vertical bar | is read "such that." This definition works because there is an understanding that the set contains every p / q that satisfies the condition to the right of the bar. Another example, the set {1, 3, 5, 7, 9, 11, . .. } can be defined as: { n Z | n = 2k+1 for some integer k >= 0 } Some other important facts about sets: 1) Two sets are equal if and only if they have the same elements: {1, 3, 5, 5, 3, 1, 1 } = {5, 1, 3} 2) If A and B are sets, then A is a subset of B if and only if, every element of A is also an element of B: A = {1,2,5,7} B = {1, 5} C = {1, 5} B is a subset of A; B is a subset of C. Another way of defining set equality is to say A = B if and only if A is a subset of B and B is a subset of A. Note that if you need to prove set equality, you have to prove two things. 3) If A is a subset of B but A does not equal B, then A is called a proper subset . In the example above, B is a proper subset of A, but B is not a proper subset of C. 4) Venn diagrams are often used to graphically represent sets. In Venn diagrams, the Universal Set (U) contains all the objects under consideration. This is represented by a rectangle. Inside the rectangle are circles or other geometrical figures used to represent sets. Venn diagrams are often used to illustrate the relationship between sets. For example, the first diagram below shows that A is a subset of B.
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This note was uploaded on 10/01/2011 for the course CS 103B taught by Professor Sahami,m during the Winter '08 term at Stanford.

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06. Sets 1 - Maggie Johnson Handout #6 CS103B Sets Key...

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