# Sets, Venn Diagrams & Counting

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 MT 142 College Mathematics Sets, Venn Diagrams & Counting Module SC Terri Miller revised January 5, 2011 What is a set? Sets set is a collection of objects. The objects in the set are called elements of the set. set is well-defined if there is a way to determine if an object belongs to the set or not. To indicate that we are considering a set, the objects (or the description) are put inside a pair of set braces, {}. Example 1. re the following sets well-defined? (1) The set of all groups of size three that can be selected from the members of this class. (2) The set of all books written by John Grisham. (3) The set of great rap artists. (4) The best fruits. (5) The 10 top-selling recording artists of Solution: (1) You can determine if a group has three people and whether or not those people are members of this class so this is well-defined. (2) You can determine whether a book was written by John Grisham or not so this is also a well-defined set. (3) rap artist being great is a matter of opinion so there is no way to tell if a particular rap artist is in this collection, this is not well-defined. (4) Similar to the previous set, best is an opinion, so this set is not well-defined. (5) This is well-defined, the top selling recording artists of any particular year are a matter of record. Equality? Two sets are equal if they contain exactly the same elements. Example 2. (1) {1, 3, 4, 5} is equal to the set {5, 1, 4, 3} (2) The set containing the letters of the word railed is equal to the set containing the letters of the word redial. There are two basic ways to describe a set. The first is by giving a description, as we did in Example 1 and the second is by listing the elements as we did in Example 2 (1).

2 2 Sets, Venn Diagrams & Counting Notation:. We usually use an upper case letter to represent a set and a lower case x to represent a generic element of a set. The symbol is used to replace the words is an element of ; the expression x would be read as x is an element of. If two sets are equal, we use the usual equal sign: = B. Example 3. = {1, 2, 3, 5} B = {m, o, a, n} C = {x x 3 and x R} D = {persons the person is a registered Democraat} = {countries the country is a member of the nited Nations} niverse? In order to work with sets we need to define a niversal Set,, which contains all possible elements of any set we wish to consider. The niversal Set is often obvious from context but on occasion needs to be explicitly stated. For example, if we are counting objects, the niversal Set would be whole numbers. If we are spelling words, the niversal Set would be letters of the alphabet. If we are considering students enrolled in S math classes this semester, the niversal Set could be all S students enrolled this semester or it could be all S students enrolled from 2000 to In this last case, the niversal Set is not so obvious and should be clearly stated. Empty? On occasion it may turn out that a set has no elements, the set is empty. Such a set is called the empty set and the notation for the empty set is either the symbol or a set of braces alone, {}. Example 4. Suppose is the set of all integers greater than 3 and less than 1. What are the elements of? There are no numbers that meet this condition, so =. Subset. is a subset of B if every element that is in is also in B. The notation for is a subset of B is B. Note: and B can be equal. Example 5. = {0, 1, 2, 3, 4, 5} B = {1, 3, 4} C = {6, 4, 3, 1} D = {0, 1, 2, 5, 3, 4} E = {} Which of the sets B, C, D, E are subsets of? B since it s elements 1, 3, and 4 are all also in. C is NOT a subset of (C ) since there is a 6 in C and there is no 6 in. D is a subset of since everything that is in D is also in ; in fact D =. Finally, E is a subset of ; this is true since any element that IS in E is also in.

3 MT Module SC 3 Notice that every set is a subset of itself and the empty set is a subset of every set. If B and B, then we say that is a proper subset of B. The notation is only a bit different: B. Note the lack of the equal part of the wymbol. Complement. Every set is a subset of some universal set. If then the complement of is the set of all elements in that are NOT in. This is denoted:. Note that =, i.e. the complement of the complement is the original set. Example 6. Consider the same sets as in Example 5. It appears that the set of all integers, = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, would be a natural choice for the univere in this case. So, we would have = {6, 7, 8, 9} B = {0, 2, 5, 6, 7, 8, 9} C = {0, 2, 5, 7, 8, 9} E = Venn Diagrams Pictures are your friends! It is often easier to understand relationships if we have something visual. For sets we use Venn diagrams. Venn diagram is a drawing in which there is a rectangle to represent the universe and closed figures (usually circles) inside the rectangle to represent sets. One way to use the diagram is to place the elements in the diagram. To do this, we write/draw those items that are in the set inside of the circle. Those items in the universe that are not in the set go outside of the circle. Example 7. With the sets = {red, orange, yellow, green, blue, indigo, violet} and = {red, yellow, blue}, the diagram looks like

4 4 Sets, Venn Diagrams & Counting red, yellow, blue indigo, violet orange, green Notice that you can see as well. Everything in the universe not in, ={orange, indigo, violet, green}. If there are more sets, there are more circles, some pictures with more sets follow. B B B B 1 C Set Operations. We will need to be able to do some basic operations with sets. The first operation we will consider is called the union of sets. This is the set that we get when we combine the elements of two sets. The union of two sets, and B is the set containing all elements of both and B; the notation for union B is B. So if x is an element of or of B or of both, then x is an element of B. Example 8. For the sets = {bear, camel, horse, dog, cat} and B = {lion, elephant, horse, dog}, we would get B = {bear, camel, horse, dog, cat, lion, elephant}. 1 To see this using a Venn diagram, we would give each set a color. Then B would be anything in the diagram with any color. Note: =, the union of a set with its complement gives the universal set. Example 9. If we color the set with blue and the set B with orange, we see the set B as the parts of the diagram that have any color (blue, beige, orange).

5 MT Module SC 5 B The next operation that we will consider is called the intersection of sets. This is the set that we get when we look at elements that the two sets have in common. The intersection of two sets, and B is the set containing all elements that are in both and B; the notation for intersect B is B. So, if X is an element of and x is an element of B, then x is an element of B. Example 10. For the sets = {bear, camel, horse, dog, cat} and B = {lion, elephant, horse, dog}, we would get B = {horse, dog}. To see this using a Venn diagram, we would give each set a color. Then B would be anything in the diagram with both colors. Example 11. If we color the set with blue and the set B with orange, we see the set B as the part of the diagram that has both blue and orange resulting in a beige colored football shape. B Note: = {}, the intersection of a set with its complement is the empty set. We can extend these ideas to more than two sets. With three sets, the Venn diagram would look like

6 6 Sets, Venn Diagrams & Counting In this diagram, the three sets create several pieces when they intersect. I have given each piece a lower case letter while the three sets are labelled with the upper case letters, B, and C. I will describe each of the pieces in terms of the sets, B, and C. With three sets, the keys to completing the venn diagrams are the triangle pieces, t, v, w, and x. The darkest blue piece in the center, w, is the intersection of all three sets, so it is B C; that is the elements in common to all three sets, and B and C. The yellow piece t is part of the intersection of 2 of the sets, it is the elements that are in both and B but not in C, so it is B C. Similarly v, the purple piece, is the elements that are in both and C but not in B, C B. Finally, the blue piece x is the elements that are in both B and C but not in, B C. The next pieces to consider are the football shaped pieces formed by joining two of the triangle pieces. The piece composed of t and w is the elements in both and B, so it is the set B. The football formed by v and w is the elements that are in both and C, that is C. The last football, formed by x and w, is the elements in both B and C, i.e. B C. The areas to consider are the large outer pieces of each circle. The red region marked with s is the elements that are in but not in B or C, this is the set (B C). Similarly, the green region, u is the elements that are in B but not in or C, i.e. B ( C). Lastly, the blue region marked with y is the elements that are in C but not in of B, the set C ( B). The final region, z, outside of all the circles is the elements that are not in nor in B nor in C. It is the set ( B C).

7 MT Module SC 7 We will return to these ideas with a later example, but first we need a few more ideas. Size of a set. The cardinality of a set is the number of elements contained in the set and is denoted n(). Example 12. If = {egg, milk, flour, sugar, butter}, then n() = 5. Note, the empty set, {}, has no elements, so n({}) = 0. Properties. de Morgan s Laws ( B) = B, and ( B) = B I will illustrate the first law using Venn diagrams and leave the illustration of the other equation to the reader. Consider the Venn diagram with two sets using the colors blue for and orange for B, from example 9, we know that B is the part containing any color. B Figure 1. B So ( B) is the white part of the current diagram that is outside of both circles. The diagram to illustrate this is given below, the set is colored in red. Figure 2. ( B) Figure 1 and 2 demonstrate the left side of the equation ( B) = B. To demonstrate the right hand side of the equation, we will start with diagrams for and B.

8 8 Sets, Venn Diagrams & Counting Figure 3. B Here we see that everything outside of is blue and everything outside of B is green. If we lay one diagram over the other, any part containing both colors will be the intersection of those two sets, B. The result will be Figure 2. pplications. In this section, I will illustrate the use of Venn diagrams in some examples. Often we use the cardinality of a set for the Venn diagram rather than the actual objects. Example 13. Two programs were broadcast on television at the same time; one was the Big Game and the other was Ice Stars. The Nelson Ratings Company uses boxes attached to television sets to determine what shows are actually being watched. In its survey of 1000 homes at the midpoint of the broadcasts, their equipment showed that 153 households were watching both shows, 736 were watching the Big Game and 55 households were not watching either. How many households were watching only Ice Stars? What percentage of the households were not watching either broadcast? We begin by constructing a Venn diagram, we will use B for the Big Game and I for Ice Stars. Rather than entering the name of every household involved, we will put the cardinality of each set in its place within the diagram. So, since they told us that 153 households were watching both broadcasts, we know that n(b I) = 153, this number goes in the dark purple football area. We are told that 736 were watching the Big Game, n(b) = 736, since we already have 153 in that part of B that is in common with I, the remaining part of B will have = 583. This tells us that 583 households were watching only the Big Game.

9 MT Module SC 9 We are also told that 55 households were watching neither program, n(( B)) = 55, so that number goes outside of both circles. Finally, we know that the total of everything should be 1000, n() = Since only one area does not yet contain a number it must be the missing amount to add up to We add the three numbers that we have, = 791, and subtract that total from 1000, = 209, to get the number that were watching only Ice Stars. Filling in this number, we have a complete Venn diagram representing the survey. Now, we have the information needed to answer any questions about the survey results. In particular, we were asked how many households were watching only Ice Stars, we found this number to be 209. We were also asked what percentage of the households were watching only the Big Game. The number watching only the game was found to be 583, so we compute the percentage, (583/1000) 100% = 58.3%.

11 We start by writing down all of the information given. MT Module SC 11 n(s) = 73 n(w ) = 51 n(p ) = 27 n((s W P )) = 2 n(s W P ) = 10 n(s W ) = 33 n(w (S P )) = 18 n(s P W ) = 5 Look at this information to see if any of it can be entered into the diagram with no further work. It is best to start with the center if possible, and then the remainder of the trianglar pieces. Our information given, tells us that e = 10, h = 2. c = 18, and d = 5. We know that the football shape that is cyan and white is supposed to have 33 people, since 10 are accounted for in the white portion, = 23 = b.

12 12 Sets, Venn Diagrams & Counting We now have three of the four pieces that make up S and we know the total of all of the pieces of S, so we find the green piece, a, 73 ( ) = 35 a. We also know three of the four pieces that make up W, this gives us the purple piece, f, 51 ( ) = 0 = f. Now we have three of the four pieces of P and only need to find the red piece g, 27 (5+10) = 12 = g, to complete the Venn diagram and begin answering the questions.

13 MT Module SC 13 Now, to answer the questions (1) This is asking the size of our universal set, so we add all of the numbers in our diagram, = 103, thus 104 people were surveyed. (2) Those who have taken vacations t exactly two times of the year would be the triangular pieces that are the intersection of only two of the sets, this is all of the triangular pieces except the white one in the center, = 28, hence 28 people took vacations at exactly two times of the year. (3) Those people who took vacations at most one time of the year either took a vacation during exactly one of the seasons, these are the green, blue, and red pieces, or had not taken a vacation at all, the number outside of all the circles. dding these together, = 67, we get that 67 people had taken at most one vacation. (4) The number that took vacations during both summer and winter but not spring is the cyan section. Thus the percentage who took vacations only during those two seasons is (23/104) 100% = %. Counting Fundamental Counting Principle. I will introduce the first counting technique with an example. Example 15. Suppose a cafeteria offers a \$5 lunch special which includes one entree, a beverage, and a side. For the entree, you can choose to have either soup or a sandwich; the beverage choices are soda, lemonade, or milk; and the side choices are chips or cookies. How many different lunch combinations are there?

15 MT Module SC 15 0! = 1 1! = 1 2! = 2 1 3! = ! = The idea can be expressed in general as: n! = n(n 1)(n 2)(n 3) We could stop the expansion process at any point and indicate the remainder of the factorial in terms of a lower factorial. 5! = 5 4! = 5 4 3! = ! = ! = The ability to expand any factorial partially can be an aid in simplifying expressions involving more than one factorial. Example 18. Compute, by expanding and simplifying, 12! 9!. Solution: 12! 9! ! = 9! = ! 9! = = 1320 (by expanding) (cancelling like terms) Tree Diagram. nother tool that is very useful in counting is called a tree diagram. This is a picture that branches for each option in choice. Example 19. If I toss a penny and a nickel, how many possible outcomes are there? sing the symbols H for heads and T for tails, we get the following tree:

16 16 Sets, Venn Diagrams & Counting H T H T H T HH HT T H T T The first coin has a possible outcome of H or T, this is represented with the first branch. For each branch here, the second coin has the outcome of H or T, represented by the second set of branches. t the end of each branch, I have listed the result of starting at the beginning and following along the branches to the end. There are 4 ends, so the total number of possible outcomes of tossing a penny and a nickel is 4. Permutations and Combinations. Next we will consider he difference in the following tasks and learn how to count them. The first task is to take 3 books from a pile of 8 distinct books and line them up. The second task is to select 3 of the 8 distinct books. The first thing to notice is that we can distinguish each object from any other and that we cannot replace any book that has been used, this is called without replacement. Both of our tasks have this feature. Next, we note that the first task is line up the books, thus order makes a difference; {math, art, english} would look different from {art, math, english}. So for task one, ORDER MTTERS. In the second task, we simply choose a group of 3 with no arranging, so for task two, ORDER DOES NOT MTTER. This leads us to the definitions for these situations. permutation is an arrangement of objects. combination is a collection of objects. Next, we will learn how to count the number of permutations and combinations. The number of permutations (arrangements) without replacement of r objects from a group of n distinct objects is denoted P (n, r) or n P r and is calculated with the formula: P (n, r) = n! (n r)!. The number of combinations (groups) without replacement of r objects chosen from a group of n distinct objects is denoted C(n, r) or n C r and is calculated with the formula C(n, r) = Example 20. Consider the set {a, b, 5}. n! r!(n r)!.

17 MT Module SC 17 (1) How many permutations of 2 of the objects are possible? Solution: The first solution is to simply list out the permutations: ab, a5, ba, b5, 5a, 5b to see that there are a total of 6. The second solution is to use the formula where r is 2 (the number of objects being arranged( and n is 3 (the number of objects from which we are selecting those to arrange). This gives the result: 3! P (3, 2) = (3 2)! = 3! 1! = 6 1 = 6. (2) How many groups of two of the objects are there? Solution: gain, our first solution will be to list the possibilities: {a, b}, {a, 5}, {b, 5} to see that there are only three ways to choose two of the objects. The second solution is to use the formula with r = 2 (the number of objects to be chosen) and n = 3 (the number of objects from which we are choosing). This gives the result: 3! C(3, 2) = 2!(3 2)! = 3! 2! 1! = = 6 2 = 3. Example 21. freshman class consists of 40 students, 30 of which are women. The class needs to select a committee of 7 to represent them in the student senate. How many committees are possible if (1) the committee must have exactly 5 women? Solution: If the committee has exactly 5 women then it is composed of 5 women and 2 men, meaning that we must do two tasks in order to do the entire selection. So we need to choose 5 women and 2 men. (number of ways to choose 5 women) (number of ways to choose 2 men) Now, we have to choose 5 women from the 30 available and order does not matter; this is 30 C 5. We also need to choose 2 men form the 10 men available where order does not matter; this is 10 C 2. The number of ways to choose a committee with exactly 5 women is 30C 5 10 C 2 = (45) = 6, 412, 770. (2) the committee must have at least 5 women? Solution: In this case, we have more than one way to satisfy the condition, the choices are: (5 women and 2 men) or (6 women and 1 man) or (7 women). The first is the number that we computed in the previous part. sing the same idea for the other two options, we have 30C 5 10 C C 6 10 C C 7 = 14, 386, 320.

### Math 117 Chapter 7 Sets and Probability

Math 117 Chapter 7 and Probability Flathead Valley Community College Page 1 of 15 1. A set is a well-defined collection of specific objects. Each item in the set is called an element or a member. Curly

### 2.1 Symbols and Terminology

2.1 Symbols and Terminology Definitions: set is a collection of objects. The objects belonging to the set are called elements, ormembers, oftheset. Sets can be designated in one of three different ways:

### Assigning Probabilities

What is a Probability? Probabilities are numbers between 0 and 1 that indicate the likelihood of an event. Generally, the statement that the probability of hitting a target- that is being fired at- is

### SETS. Chapter Overview

Chapter 1 SETS 1.1 Overview This chapter deals with the concept of a set, operations on sets.concept of sets will be useful in studying the relations and functions. 1.1.1 Set and their representations

### 3.1 Events, Sample Spaces, and Probability

University of California, Davis Department of Statistics Summer Session II Statistics 13 August 6, 2012 Lecture 3: Probability 3.1 Events, Sample Spaces, and Probability Date of latest update: August 8

### Probability. Vocabulary

MAT 142 College Mathematics Probability Module #PM Terri L. Miller & Elizabeth E. K. Jones revised January 5, 2011 Vocabulary In order to discuss probability we will need a fair bit of vocabulary. Probability

### A1. Basic Reviews PERMUTATIONS and COMBINATIONS... or HOW TO COUNT

A1. Basic Reviews Appendix / A1. Basic Reviews / Perms & Combos-1 PERMUTATIONS and COMBINATIONS... or HOW TO COUNT Question 1: Suppose we wish to arrange n 5 people {a, b, c, d, e}, standing side by side,

### Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments

### The Mathematics Driving License for Computer Science- CS10410

The Mathematics Driving License for Computer Science- CS10410 Venn Diagram, Union, Intersection, Difference, Complement, Disjoint, Subset and Power Set Nitin Naik Department of Computer Science Venn-Euler

### Chapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.

MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.

### 11.5. A summary of this principle is now stated. The complement of a set FIGURE 8

11.5 The complement of a set FIGRE 8 Counting Problems Involving Not and Or When counting the number of ways that an event can occur (or that a task can be done), it is sometimes easier to take an indirect

### 7.5 Conditional Probability; Independent Events

7.5 Conditional Probability; Independent Events Conditional Probability Example 1. Suppose there are two boxes, A and B containing some red and blue stones. The following table gives the number of stones

### 94 Counting Solutions for Chapter 3. Section 3.2

94 Counting 3.11 Solutions for Chapter 3 Section 3.2 1. Consider lists made from the letters T, H, E, O, R, Y, with repetition allowed. (a How many length-4 lists are there? Answer: 6 6 6 6 = 1296. (b

### Sets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.

Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, complex numbers, C integers, All students in

### What is a set? Sets. Specifying a Set. Notes. The Universal Set. Specifying a Set 10/29/13

What is a set? Sets CS 231 Dianna Xu set is a group of objects People: {lice, ob, Clara} Colors of a rainbow: {red, orange, yellow, green, blue, purple} States in the S: {labama, laska, Virginia, } ll

### (Refer Slide Time 1.50)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Module -2 Lecture #11 Induction Today we shall consider proof

Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish

### Lecture 11: Probability models

Lecture 11: Probability models Probability is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model we need the following ingredients A sample

### not to be republishe NCERT SETS Chapter Introduction 1.2 Sets and their Representations

SETS Chapter 1 In these days of conflict between ancient and modern studies; there must surely be something to be said for a study which did not begin with Pythagoras and will not end with Einstein; but

### Sections 2.1, 2.2 and 2.4

SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete

### Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting

Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Colin Stirling Informatics Slides originally by Kousha Etessami Colin Stirling (Informatics) Discrete Mathematics (Chapter 6) Today 1 /

### Announcements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets

CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.3-2.6 Homework 2 due Tuesday Recitation 3 on Friday

### Chapter 5 - Probability

Chapter 5 - Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set

### 4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.

Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,

### Hawkes Learning Systems: College Algebra

Hawkes Learning Systems: College Algebra Section 1.2: The Arithmetic of Algebraic Expressions Objectives o Components and terminology of algebraic expressions. o The field properties and their use in algebra.

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

### Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### 4.3. Addition and Multiplication Laws of Probability. Introduction. Prerequisites. Learning Outcomes. Learning Style

Addition and Multiplication Laws of Probability 4.3 Introduction When we require the probability of two events occurring simultaneously or the probability of one or the other or both of two events occurring

### 33 Probability: Some Basic Terms

33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been

### 1 Introduction. 2 Basic Principles. 2.1 Multiplication Rule. [Ch 9] Counting Methods. 400 lecture note #9

400 lecture note #9 [Ch 9] Counting Methods 1 Introduction In many discrete problems, we are confronted with the problem of counting. Here we develop tools which help us counting. Examples: o [9.1.2 (p.

### Methods Used for Counting

COUNTING METHODS From our preliminary work in probability, we often found ourselves wondering how many different scenarios there were in a given situation. In the beginning of that chapter, we merely tried

### Basic concepts in probability. Sue Gordon

Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

### Activity Symbol per Hour 2.2 Venn Diagrams and Subsets universe of discourse. universal set. Venn diagrams, F I G U R E 1

U 2.2 A A FIGURE 1 Venn Diagrams and Subsets In every problem there is either a stated or implied universe of discourse. The universe of discourse includes all things under discussion at a given time.

### Math 421: Probability and Statistics I Note Set 2

Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the

### CHAPTER 2. Set, Whole Numbers, and Numeration

CHAPTER 2 Set, Whole Numbers, and Numeration 2.1. Sets as a Basis for Whole Numbers A set is a collection of objects, called the elements or members of the set. Three common ways to define sets: (1) A

### Elements of probability theory

2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted

### Sets and set operations

CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used

### Exam 1 Review Math 118 All Sections

Exam Review Math 8 All Sections This exam will cover sections.-.6 and 2.-2.3 of the textbook. No books, notes, calculators or other aids are allowed on this exam. There is no time limit. It will consist

### Check Skills You ll Need. New Vocabulary union intersection disjoint sets. Union of Sets

NY-4 nion and Intersection of Sets Learning Standards for Mathematics..31 Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets). Check Skills You ll Need

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### SECTION 10-5 Multiplication Principle, Permutations, and Combinations

10-5 Multiplication Principle, Permutations, and Combinations 761 54. Can you guess what the next two rows in Pascal s triangle, shown at right, are? Compare the numbers in the triangle with the binomial

### I. WHAT IS PROBABILITY?

C HAPTER 3 PROBABILITY Random Experiments I. WHAT IS PROBABILITY? The weatherman on 0 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

### 34 Probability and Counting Techniques

34 Probability and Counting Techniques If you recall that the classical probability of an event E S is given by P (E) = n(e) n(s) where n(e) and n(s) denote the number of elements of E and S respectively.

### 1 Introduction to Counting

1 Introduction to Counting 1.1 Introduction In this chapter you will learn the fundamentals of enumerative combinatorics, the branch of mathematics concerned with counting. While enumeration problems can

### Sequences and Series

Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will

### EXAM. Exam #3. Math 1430, Spring 2002. April 21, 2001 ANSWERS

EXAM Exam #3 Math 1430, Spring 2002 April 21, 2001 ANSWERS i 60 pts. Problem 1. A city has two newspapers, the Gazette and the Journal. In a survey of 1, 200 residents, 500 read the Journal, 700 read the

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

### 4.1. Sets. Introduction. Prerequisites. Learning Outcomes. Learning Style

ets 4.1 Introduction If we can identify a property which is common to several objects, it is often useful to group them together. uch a grouping is called a set. Engineers for example, may wish to study

### A Little Set Theory (Never Hurt Anybody)

A Little Set Theory (Never Hurt Anybody) Matthew Saltzman Department of Mathematical Sciences Clemson University Draft: August 21, 2013 1 Introduction The fundamental ideas of set theory and the algebra

### Clicker Question. Theorems/Proofs and Computational Problems/Algorithms MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES

MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Tuesday, 1/21/14 General course Information Sets Reading: [J] 1.1 Optional: [H] 1.1-1.7 Exercises: Do before next class; not to hand in [J] pp. 12-14:

### LESSON SUMMARY. Set Operations and Venn Diagrams

LESSON SUMMARY CXC CSEC MATHEMATICS UNIT Three: Set Theory Lesson 4 Set Operations and Venn Diagrams Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volumes 1 and 2. (Some helpful exercises

### Chapter 4: Probabilities and Proportions

Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 4: Probabilities and Proportions Section 4.1 Introduction In the real world,

### SAT Mathematics. Numbers and Operations. John L. Lehet

SAT Mathematics Review Numbers and Operations John L. Lehet jlehet@mathmaverick.com www.mathmaverick.com Properties of Integers Arithmetic Word Problems Number Lines Squares and Square Roots Fractions

### COUNTING SUBSETS OF A SET: COMBINATIONS

COUNTING SUBSETS OF A SET: COMBINATIONS DEFINITION 1: Let n, r be nonnegative integers with r n. An r-combination of a set of n elements is a subset of r of the n elements. EXAMPLE 1: Let S {a, b, c, d}.

### Set operations and Venn Diagrams. COPYRIGHT 2006 by LAVON B. PAGE

Set operations and Venn Diagrams Set operations and Venn diagrams! = { x x " and x " } This is the intersection of and. # = { x x " or x " } This is the union of and. n element of! belongs to both and,

### All the examples in this worksheet and all the answers to questions are available as answer sheets or videos.

BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Numbers 3 In this section we will look at - improper fractions and mixed fractions - multiplying and dividing fractions - what decimals mean and exponents

### 2015 Junior Certificate Higher Level Official Sample Paper 1

2015 Junior Certificate Higher Level Official Sample Paper 1 Question 1 (Suggested maximum time: 5 minutes) The sets U, P, Q, and R are shown in the Venn diagram below. (a) Use the Venn diagram to list

### 36 Odds, Expected Value, and Conditional Probability

36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### SOLUTIONS, Homework 6

SOLUTIONS, Homework 6 1. Testing out the identities from Section 6.4. (a) Write out the numbers in the first nine rows of Pascal s triangle. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

### 1. The sample space S is the set of all possible outcomes. 2. An event is a set of one or more outcomes for an experiment. It is a sub set of S.

1 Probability Theory 1.1 Experiment, Outcomes, Sample Space Example 1 n psychologist examined the response of people standing in line at a copying machines. Student volunteers approached the person first

### 1. De Morgan s Law. Let U be a set and consider the following logical statement depending on an integer n. We will call this statement P (n):

Math 309 Fall 0 Homework 5 Drew Armstrong. De Morgan s Law. Let U be a set and consider the following logical statement depending on an integer n. We will call this statement P (n): For any n sets A, A,...,

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Access The Mathematics of Internet Search Engines

Lesson1 Access The Mathematics of Internet Search Engines You are living in the midst of an ongoing revolution in information processing and telecommunications. Telephones, televisions, and computers are

### Math 166 - Week in Review #4. A proposition, or statement, is a declarative sentence that can be classified as either true or false, but not both.

Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Week in Review #4 Sections A.1 and A.2 - Propositions, Connectives, and Truth Tables A proposition, or statement, is a declarative sentence that

### Pigeonhole Principle Solutions

Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such

### Standards for Logical Reasoning

Standards for Logical Reasoning General Outcome Develop logical reasoning. General Notes: This topic is intended to develop students numerical and logical reasoning skills, which are applicable to many

### 16.1. Sequences and Series. Introduction. Prerequisites. Learning Outcomes. Learning Style

Sequences and Series 16.1 Introduction In this block we develop the ground work for later blocks on infinite series and on power series. We begin with simple sequences of numbers and with finite series

### NOTES ON COUNTING KARL PETERSEN

NOTES ON COUNTING KARL PETERSEN It is important to be able to count exactly the number of elements in any finite set. We will see many applications of counting as we proceed (number of Enigma plugboard

### https://assessment.casa.uh.edu/assessment/printtest.htm PRINTABLE VERSION Quiz 10

1 of 8 4/9/2013 8:17 AM PRINTABLE VERSION Quiz 10 Question 1 Let A and B be events in a sample space S such that P(A) = 0.34, P(B) = 0.39 and P(A B) = 0.19. Find P(A B). a) 0.4872 b) 0.5588 c) 0.0256 d)

### Name Date. Goal: Understand and represent the intersection and union of two sets.

F Math 12 3.3 Intersection and Union of Two Sets p. 162 Name Date Goal: Understand and represent the intersection and union of two sets. A. intersection: The set of elements that are common to two or more

### Equivalence relations

Equivalence relations A motivating example for equivalence relations is the problem of constructing the rational numbers. A rational number is the same thing as a fraction a/b, a, b Z and b 0, and hence

### Section 6-5 Sample Spaces and Probability

492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)

### Discrete probability and the laws of chance

Chapter 8 Discrete probability and the laws of chance 8.1 Introduction In this chapter we lay the groundwork for calculations and rules governing simple discrete probabilities. These steps will be essential

### THE LANGUAGE OF SETS AND SET NOTATION

THE LNGGE OF SETS ND SET NOTTION Mathematics is often referred to as a language with its own vocabulary and rules of grammar; one of the basic building blocks of the language of mathematics is the language

### Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

### Math 1320 Chapter Seven Pack. Section 7.1 Sample Spaces and Events. Experiments, Outcomes, and Sample Spaces. Events. Complement of an Event

Math 1320 Chapter Seven Pack Section 7.1 Sample Spaces and Events Experiments, Outcomes, and Sample Spaces An experiment is an occurrence with a result, or outcome, that is uncertain before the experiment

### Discrete Mathematics: Homework 6 Due:

Discrete Mathematics: Homework 6 Due: 2011.05.20 1. (3%) How many bit strings are there of length six or less? We use the sum rule, adding the number of bit strings of each length to 6. If we include the

### Consumer Math 15 INDEPENDENT LEAR NING S INC E 1975. Consumer Math

Consumer Math 15 INDEPENDENT LEAR NING S INC E 1975 Consumer Math Consumer Math ENROLLED STUDENTS ONLY This course is designed for the student who is challenged by abstract forms of higher This math. course

### MAT 1000. Mathematics in Today's World

MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities

### Section 2.1. Tree Diagrams

Section 2.1 Tree Diagrams Example 2.1 Problem For the resistors of Example 1.16, we used A to denote the event that a randomly chosen resistor is within 50 Ω of the nominal value. This could mean acceptable.

### Lectures 2 and 3 : Probability

Lectures 2 and 3 : Probability Jonathan L. Marchini October 15, 2004 In this lecture we will learn about why we need to learn about probability what probability is how to assign probabilities how to manipulate

### (Refer Slide Time: 1:41)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

### Lecture 1. Basic Concepts of Set Theory, Functions and Relations

September 7, 2005 p. 1 Lecture 1. Basic Concepts of Set Theory, Functions and Relations 0. Preliminaries...1 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### Compound Inequalities. Section 3-6

Compound Inequalities Section 3-6 Goals Goal To solve and graph inequalities containing the word and. To solve and graph inequalities containing the word or. Vocabulary Compound Inequality Interval Notation

### Sets and Logic. Chapter Sets

Chapter 2 Sets and Logic This chapter introduces sets. In it we study the structure on subsets of a set, operations on subsets, the relations of inclusion and equality on sets, and the close connection

### CmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B

CmSc 175 Discrete Mathematics Lesson 10: SETS Sets: finite, infinite, : empty set, U : universal set Describing a set: Enumeration = {a, b, c} Predicates = {x P(x)} Recursive definition, e.g. sequences

### Probability. Experiment - any happening for which the result is uncertain. Outcome the possible result of the experiment

Probability Definitions: Experiment - any happening for which the result is uncertain Outcome the possible result of the experiment Sample space the set of all possible outcomes of the experiment Event

### Notes on counting finite sets

Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................

### Year 8 - Maths Autumn Term

Year 8 - Maths Autumn Term Whole Numbers and Decimals Order, add and subtract negative numbers. Recognise and use multiples and factors. Use divisibility tests. Recognise prime numbers. Find square numbers

### PROBABILITY. Chapter Overview

Chapter 6 PROBABILITY 6. Overview Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability

### MOL # 1 1. Make a table of the days of the week & # the days starting with 1 day from today in the problem. Now expand that out. You do not have to #

MOL # 1 1. Make a table of the days of the week & # the days starting with 1 day from today in the problem. Now expand that out. You do not have to # everything out to 100.. Think of counting by.. 2. Make

### Spring 2007 Math 510 Hints for practice problems

Spring 2007 Math 510 Hints for practice problems Section 1 Imagine a prison consisting of 4 cells arranged like the squares of an -chessboard There are doors between all adjacent cells A prisoner in one

### MATH 214 Mathematical Problem Solving for MCED Spring 2011

MATH 214 Mathematical Problem Solving for MCED Spring 2011 Venn Diagrams 1 The English logician John Venn (1834-1923) 2 invented a simple visual way of describing relationships between sets. His diagrams,