Slide 1 Math 1520, Lecture 13

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1 Slide 1 Math 1520, Lecture 13 In chapter 7, we discuss background leading up to probability. Probability is one of the most commonly used pieces of mathematics in the world. Understanding the basic concepts of sets and counting is fundamental to understanding probability. In this lecture we discuss some useful formulas that can be used to count the number of elements in a set. This is part of a much larger subject called combinatorics.

2 Slide 2 Set Notation and Terminology (Review) 1. A set is an unordered collection of objects so that no object may occur twice and we can determine definitively whether an object is an element of a set or not. We use braces, { }, to denote a set. Example: A = {a, b, c} 2. The objects in a set are called elements. If a is an element of a set A we write a A. Example: b {a, b, c} 3. A set written in roster notation lists all the elements of the set. Example: B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 4. A set written in set builder notation gives a rule that describes the property or properties of the set. Example: B = {x x is an integer between 1 and 10} 5. Two sets are equal if they have exactly the same elements. Example: {a, b, c} = {b, c, a} 6. A is a subset of B if every element of A is also an element of B. We write this as A B. Example: {x x is a maple tree} {x x is a tree} 7. A is a proper subset of B if every element of A is also an element of B but A B. We write this as A B. Example: {x x is a maple tree} {x x is a tree} 8. The empty set is written as and is the set that has no elements. It is a subset of every set. Example: {x x is a tree} 9. The universal set is written as U and is the set that contains all objects of interest in a particular application. It is context dependent. Example: U is the set of all possible 5 card poker hands if we are studying only 5 card poker hands.

3 Slide 3 Unions, Intersections and Complements (Review) We frequently use the following operations on sets. The union of two sets A and B is written A B and is the collection of all elements that are either in A or in B or both. Example: {1, 3, 4} {1, 2, 4, 5} = {1, 2, 3, 4, 5} The intersection of two sets A and B is written A B and is the collection of all elements that are in both A and B. Example: {1, 3, 4} {1, 2, 4, 5} = {1, 4} Two sets are disjoint if their intersection is empty. Example: {1} {2} = so {1} and {2} are disjoint sets. The complement of a set A in the universal set U is written A c and is the set of all elements that are in U but not in A. Example: If U = {1, 2, 3, 4, 5} and A = {1, 3, 4} then A c = {2, 5}.

4 Slide 4 Why do we need counting in Probability? Recall this example from the last lecture: What is the probability of getting a flush in poker? We could write the set of all flushes to be F = S D H C where S = {x : x is a poker hand consisting of 5 spades} D = {x : x is a poker hand consisting of 5 diamonds} H = {x : x is a poker hand consisting of 5 hearts} C = {x : x is a poker hand consisting of all clubs} and the symbol means to take the union. 1. In finding the probability, we need to count the number of ways we get a flush. 2. To do this we need to count how many elements are in each of the sets S, D, H, C and add them together. 3. This is a basic rule we can use in general that we will see.

5 Slide 5 Counting the Number of Elements in a set The number of elements in the finite set A is denoted by n(a). This is also sometimes called the cardinality of the set. (All the sets for this lecture will be finite.) Examples 1. n( ) = 0 2. n({a, b, c, d}) = 4 3. If A = {1, 2, 3} and B = {4, 5} then n(a B) = 5 4. If A = {a, b, c} and B = {c, a, z} then n(a B) = 4 5. If A = {a, b, c} and B = {c, a, z} then n(a B) = 2 6. If U = {x x is an integer with 1 x 10} and A = {2, 4, 8} then n(a c ) = 7

6 Slide 6 iclicker Question If A = {1, 2, 3, 5, 7} and B = {2, 4} then what is n(a B) A. 6 B. 5 C. 7 D. 4

7 Answer to Question If A = {1, 2, 3, 5, 7} and B = {2, 4} then what is n(a B) A. 6 is the correct answer. B. 5 C. 7 D. 4

8 Slide 7 Counting Rules On previous slides we encountered problems on finding the number of elements in sets with unions. Here are some general rules that may help. If U is the universal set and A U then n(a c ) = n(u) n(a) If A and B are disjoint sets then n(a B) = n(a) + n(b) If A and B are two sets, not necessarily disjoint, then n(a B) = n(a) + n(b) n(a B) If A, B, C are three sets, not necessarily disjoint, then n(a B C) = n(a) + n(b) + n(c) n(a B) n(a C) n(b C) +n(a B C) This can be generalized to any (finite) number of (finite) sets. 1. Illustrate these rules using Venn Diagrams and/or examples.

9 Slide 8 iclicker Question Suppose that n(b) = 10, n(a B) = 4 and n(a) = 12 then what is n(a B) A. 22 B. 26 C. 14 D. 16

10 Answer to Question Suppose that n(b) = 10, n(a B) = 4 and n(a) = 12 then what is n(a B) A. 22 B. 26 C. 14 D. 16 is the correct answer. The correct answer is = 18

11 Slide 9 iclicker Question Suppose that n(b) = 5, n(a B) = 6 and n(a B) = 3 then what is n(a) A. 4 B. 3 C. 5 D. 7

12 Answer to Question Suppose that n(b) = 5, n(a B) = 6 and n(a B) = 3 then what is n(a) A. 4 is the correct answer. B. 3 C. 5 D. 7 Substituting the known numbers into n(a B) = n(a) + n(b) n(a B) we get 6 = n(a) so we solve for n(a) to get n(a) = 4.

13 Slide 10 iclicker Question Of 50 employees of a store located in downtown Boston, 18 people take the subway to work, 12 take the bus and 7 take both the subway and the bus. How many take either the subway or the bus or both to work? A. 23 B. 50 C. 19 D. 30

14 Answer to Question Of 50 employees of a store located in downtown Boston, 18 people take the subway to work, 12 take the bus and 7 take both the subway and the bus. How many take either the subway or the bus or both to work? A. 23 is the correct answer. B. 50 C. 19 D. 30 The answer is = 23

15 Slide 11 iclicker Question Of 50 employees of a store located in downtown Boston, 18 people take the subway to work, 12 take the bus and 7 take both the subway and the bus. How many come to work without taking the subway or the bus? A. 23 B. 27 C. 20 D. 13

16 Answer to Question Of 50 employees of a store located in downtown Boston, 18 people take the subway to work, 12 take the bus and 7 take both the subway and the bus. How many come to work without taking the subway or the bus? A. 23 B. 27 is the correct answer. C. 20 D. 13 If U is the set of all employees, we want n ((A B) c ) where A is the set of people that take the subway and B is the set of people that take the bus. We calculate n ((A B) c ) = n(u) n(a B) = = 27

17 Slide 12 iclicker Question Of 50 employees of a store located in downtown Boston, 18 people take the subway to work, 12 take the bus and 7 take both the subway and the bus. How many people take only the subway (and not the bus) to work? A. 11 B. 5 C. 12 D. 7

18 Answer to Question Of 50 employees of a store located in downtown Boston, 18 people take the subway to work, 12 take the bus and 7 take both the subway and the bus. How many people take only the subway (and not the bus) to work? A. 11 is the correct answer. B. 5 C. 12 D. 7 The answer is 18 7 = 11

19 Slide 13 iclicker Question To help plan the number of meals to be prepared at a college cafeteria, a survey conducted of students revealed the following data: 130 students ate breakfast, 180 students ate lunch, 275 students ate dinner, and 68 ate breakfast and lunch, 112 ate breakfast and dinner, 90 ate lunch and dinner, and 58 students ate all three meals. How many students were there that ate at least one meal in the cafeteria according to the survey? A. 370 B. 371 C. 372 D. 373

20 Answer to Question To help plan the number of meals to be prepared at a college cafeteria, a survey conducted of students revealed the following data: 130 students ate breakfast, 180 students ate lunch, 275 students ate dinner, and 68 ate breakfast and lunch, 112 ate breakfast and dinner, 90 ate lunch and dinner, and 58 students ate all three meals. How many students were there that ate at least one meal in the cafeteria according to the survey? A. 370 B. 371 C. 372 D. 373 is the correct answer. The total is = 373

21 Slide 14 iclicker Question To help plan the number of meals to be prepared at a college cafeteria, a survey conducted of students revealed the following data: 130 students ate breakfast, 180 students ate lunch, 275 students ate dinner, and 68 ate breakfast and lunch, 112 ate breakfast and dinner, 90 ate lunch and dinner, and 58 students ate all three meals. How many students were there that ate at exactly two meals in the cafeteria according to the survey? A. 99 B. 98 C. 97 D. 96

22 Answer to Question To help plan the number of meals to be prepared at a college cafeteria, a survey conducted of students revealed the following data: 130 students ate breakfast, 180 students ate lunch, 275 students ate dinner, and 68 ate breakfast and lunch, 112 ate breakfast and dinner, 90 ate lunch and dinner, and 58 students ate all three meals. How many students were there that ate at exactly two meals in the cafeteria according to the survey? A. 99 B. 98 C. 97 D. 96 is the correct answer. This is an example where it is best to draw out a Venn Diagram rather than using formulas.

Section Introduction to Sets

Section Introduction to Sets Section 1.1 - Introduction to Sets Definition: A set is a well-defined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase