2.1 Symbols and Terminology

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1 2.1 Symbols and Terminology A is a collection of objects or things. The objects belonging to the are called the, or. - : there is a way of determining for sure whether a particular item is an element of the set. Ex 1) The set of all movies directed by Stephen Spielberg. Ex 2) The set of all great restaurants. Writing Sets: a) b) c) Ex 3) Describe the following in words and in roster notation: states the state begins with the letter a Note: We don t list an element more than once in a given set. We write it numerically or alphabetically. The set containing no elements is called the or. Ex 4) Give a complete listing of all the elements of each of the following sets. a) The set of counting numbers between seventeen and twenty three. b) {x x is an even whole number less than 11} Cosner - Math 107 Chapter 2 Notes - 1

2 means means Ex 5) Fill in the blank to make the statement true. a) If A = {1, 2, 3}, then 1 A. b) A = {1, 2, 3}, then 17 A. Sets of Numbers Natural or Counting Numbers Whole Numbers Integers Rational Numbers Real Numbers Irrational Numbers Cardinal number: n A of set A is the in the set. Ex 6) Find the cardinal number of each set. a) A = {1, 2, 3} b) M = {23} c) P = { } Finite Set: A set is finite if the of a set is a particular whole number. Otherwise the set is said to be. Ex 7) What is an example of an infinite set? Equal sets: Two sets are equal if the contain exactly the same elements. Ex 8) Determine whether the statement is true or false: {x x is a whole number less than 3} = {1, 2}. Cosner - Math 107 Chapter 2 Notes - 2

3 2.2 Venn Diagrams and Subsets There is a stated or implied in the statement of a problem. This includes all things under discussion at a given time. In set theory, this is called a. We use Venn diagrams to illustrate the universal set, represented using a rectangle, and other sets depicted by circular regions. The Complement of a Set: For any set A within the universal set U, the complement of A, written A ', is the set of elements of U that are not elements of A. Ex 1) Given U = {1, 2, 3,,10}, A = {1,2, 3, 4}, B={1,2, 3, 4, 5,6}. Find each of the following sets: a) B b) A c) U Subset of a Set: Set A is a subset of set B if every element of A is also an element of B. Set Equality: Suppose A and B are sets. Then A = B if Proper Subset of a Set: Set A is a proper subset of set B if A B and A B. Ex 2) U = {1,2, 3, 10} A = {1,2, 3, 5} B = {1} C = {1, 2, 3, 4} Identify subsets and proper subsets. Cosner - Math 107 Chapter 2 Notes - 3

4 Ex 3) Are the following true always, sometimes or never? a) b) Ex 4) Find all possible subsets of each set. c) B = {1} d) C = {1, 2, 3} e) D = {0, 4, 7, 9} Number of Subsets: The number of subsets of a set with n elements is Number of Proper Subsets: The number of proper subsets of a set with n elements is Ex 5) Find the number of subsets and the number of proper subsets of {4, 6, 8, 17, 23} Cosner - Math 107 Chapter 2 Notes - 4

5 2.3 Set Operations and Cartesian Products Intersection of Sets: The of sets A and B, written by A B, is the set of elements common to both A and B, or A B Union of Sets: The of sets A and B, written by elements belonging to either of the sets, or AU B A B, is the set of all Ex 1) Given U = {1, 2, 3,,10}, A = {1,2, 3, 4}, B={1,2, 3, 4, 5,6}. Find each of the following sets: d) A U B e) A B f) A U U g) U B h) A U Ø i) B Ø Cosner - Math 107 Chapter 2 Notes - 5

6 Disjoint Sets: Sets with in common are called disjoint sets. Ex 2) What is an example of two disjoint sets? Ex 3) Let U = {q, s, u, v, w, x, y, z}, A = {q, s, y, z}, B = {v, w, x, y, z}, and C = {s}. Find each set. a) A B b) B C c) A (B C ) d) (A C ) B Cosner - Math 107 Chapter 2 Notes - 6

7 Ex 4) Describe each of the following sets in words. c) A (B C ) d) (A C ) B Difference of Sets: The of sets A and B, written A - B, is the set of elements belonging to set A and to set B, or A B Ex 5) Given U = {1, 2, 3, 4, 5, 6, 7}, A = {1, 2, 3, 4, 5, 6}, B={2, 3, 6}, C = {3, 5, 7}. Find each set: a) A - B b) B - A c) (A B) U C Cosner - Math 107 Chapter 2 Notes - 7

8 Ordered Pairs: In the ordered pair (a, b), a is called the component and b is called the component. In general, (a, b) (b, a) Cartesian Product: The Cartesian product of sets A and B, written A B A B, is Ex 6) Let A = {4, 6, 8} and B = {0, 5}. Find each set. a) A B b) B A c) A A Cardinal Number of a Cartesian Product: If n(a) = a, and n(b) = b, then n A B Ex 7) Draw a Venn Diagram and shade the region or regions representing the following sets: a) A' B b) B' U A' Cosner - Math 107 Chapter 2 Notes - 8

9 c) A' B' C d) (A B) e) A B De Morgan s Laws: For any sets A and B, Ex 8) Write a description of the shaded area. a) b) Cosner - Math 107 Chapter 2 Notes - 9

10 2.4 Surveys and Cardinal Numbers Problems involving sets of people (or other objects) sometimes require analyzing known information about certain subsets to obtain cardinal numbers of other subsets. The known information is often obtained by administering a survey. Venn diagrams and the formulas help organize the data. Ex 1) Department Store Survey 428 shoppers 214 made a purchase 299 satisfied with service 52 who made a purchase were not satisfied with service a) How many shoppers made a purchase and were satisfied with purchase? b) How many shoppers made a purchase or were satisfied with service? c) How many shoppers were satisfied with service but did not make a purchase? d) How many shoppers were not satisfied and did not make a purchase? Cardinal Number Formula for the Union of Sets: n( AU B) Ex 2) Suppose n ( U) 280, n ( A) 56, n ( B) 192 a) What if n( AU B) 135, find n( A B). b) What if n ( A B) 28, find n( AU B). c) What if n( AU B) 148, find n( A B). Cosner - Math 107 Chapter 2 Notes - 10

11 Ex 3) Suppose that a group of 140 people were questioned about particular sports that they watch regularly and the following information was produced. 93 like football 40 like football and baseball 20 like all three 70 like baseball 25 like baseball and hockey 40 like hockey 28 like football and hockey a) How many people like only football? b) How many people don t like any of the sports? Ex 4) On a given day, breakfast patrons were categorized according to age and preferred beverage. The results are summarized below. Coffee (C) Juice (J) Tea (T) Totals (Y) (M) Over 33 (O) Totals Using the letters in the table, find the number of people in each of the following sets. a) Y C b) O T Cosner - Math 107 Chapter 2 Notes - 11

2. Sets. 2.1&2.2: Sets and Subsets. Combining Sets. c Dr Oksana Shatalov, Fall

2. Sets. 2.1&2.2: Sets and Subsets. Combining Sets. c Dr Oksana Shatalov, Fall c Dr Oksana Shatalov, Fall 2014 1 2. Sets 2.1&2.2: Sets and Subsets. Combining Sets. Set Terminology and Notation DEFINITIONS: Set is well-defined collection of objects. Elements are objects or members