08-26-2009Sets and Operations on SetsWhat is a set?• A set is a collection of objects.• An object in a set is called an element.• Examples: Colors in the rainbow, people in the class, animals in the zoo. The numbers 1,2,3,.... Theletters in the alphab et.• When we talk ab out sets we must define the universe. The unive rse is all the objects that are allowedto be considered. For example, if P is the set of p eople in the class, the universe is the set of all possiblepeople in the world.• Well defined: It only makes sense to talk about a set when both (a) there is universe for the set and(b) every object in the universe is either in the set or not in the set. Example: Is the collection of“Best Songs of the 1980’s” a set? How can you change the definition to make it a set?How to describe a set.1. Verbal description: ”The set of people on student government.”2. Listing in braces: {monkey, giraffe, kangaroo, meerkat.}.3. Set builder notation: {x| xis a letter in the alphabet}.Note on listing sets: The order in which we list the elements in a set does not matter. So the set {x, y, z}is the same as the set {z, y, x}. Also, we only list each element of a set once so the s ets {1, 1, 1, 2, 3} and{1, 2, 3} are the same.Venn Diagrams You can use Venn diagrams to model sets.Some more definitions for sets.• If we want to s ay x is an element of a set A, we will typically write x ∈ A where the symbol “∈” means“in”.• A natural number or counting number is a member of the set N = {1, 2, 3, ...}.• The complement of a set A is the set of elements in the universe set U that are not in the set A.We will write¯A for the complement of the set A. Example: What is the complement of people inthe class? What is the complement of the set {a, b, c}? What is the complement of a universal set?Examples 2.2 on p. 82.• The empty set is the set with no elements in it. We write it as {} or ∅.• The set A is subset of B if every e lement of A is also an element of B. If A is a subset of B we willwrite A ⊂ B. Example: the set of students in the class is a subset of the people at UK which is asubset of the people currently in Kentucky.• The union of the sets A and B is the set which contains all the elements of both A and B. The unionof A and B is a set and we write it as A ∪ B. In set builder:A ∪ B = {x|x ∈ A or x ∈ B}.Example.: {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}• The intersection of the sets A and B is the set which contains all the elements which A and B havein common. The intersection of A and B is a set and we name it A ∩ B. In set builder:A ∩ B = {x|x ∈ A and x ∈ B}.Example.: {1, 2, 3} ∩ {3, 4, 5} = {3}• Two sets A and B are c alled disjoint if they have no elements in common. This is equivalent toA ∩ B = ∅.• For practice on these concepts look at the elements of the set of examples {2.2, 2.3, 2.4} in your text.Activity: Set Skits1. Class is divided into groups of 5.2. Each person in the group gets one of 5 different colored stickers. Say: red, green, blue, orange, yellow.3. Each group gets a card with several set operations in list notation. Example: {red, blue}∪{yellow, orange}.4. Groups discuss their op erations.5. Each group presents their operations to the class by forming the result of the operation at the frontof the room. Example: if the students have {red, blue} ∪ {yellow, orange}, they should make theformation {red, blue, yellow, orange} at the front of the
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