Guided Notes - Minnesota State University Moorhead
Guided Notes
to Accompany Text:
Mathematics All Around
(Text by Pirnot)
Prepared by
Dr. Kris Montis
MSUM Department of Mathematics
For
MSUM Course MA 102
Fall 2006
| |
|Problem Solving Strategies (Section 1.1 – 1.2) |
|1. Explain in your own words how each of the following contributes to the problem solving process. If possible, relate each idea to your own |
|experiences in problem solving. |
| |
|Preparation: |
| |
| |
| |
| |
| |
| |
| |
| |
|Thinking Time: |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Insight: |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Verification: |
| |
| |
| |
| |
| |
| |
| |
| |
|2. Demonstrate each strategy using the given problem or example. |
| |
|(a) Strategy: Draw a Picture. Problems usually contain several conditions that must be satisfied. You will find it useful to draw pictures to |
|understand these conditions before trying to solve the problem. |
| |
|Apply to Problem: Five liters of 10% sugar solution are mixed with pure water to get a 5% solution. Represent this situation with a picture. (Do|
|not solve, just represent with a picture). |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|(b) Strategy: Choose Good Names for Unknowns. It is a good practice to name the objects in a problem so you can remember their meanings easily. |
|Conversely, when given variable names, be sure you understand what each one stands for. |
| |
|Example: Let s represent “I will study”, t represent “I will watch TV”, and k represent “I will snack”. |
| |
|What is helpful about these variable names? |
| |
| |
| |
| |
| |
|What might be confusing if you were not paying careful attention |
| |
| |
| |
| |
| |
|Suggest a strategy you could use so you won’t confuse these variable names. |
| |
| |
| |
| |
|(c) Strategy: Be Systematic. If you approach a situation in an organized, systematic way, frequently you will gain insight into the problem. |
|Insight often comes from identifying patterns. |
| |
|Example: Make a systematic list of all the “2-letter code words” that can be made from the letters a, y, p if each letter can be used only once. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|How can your organization method help you determine how many items should be in your list? |
| |
| |
| |
| |
| |
| |
|(d) Strategy: Look for Patterns. If you can recognize a pattern in a situation you are studying you can often use it to answer questions about |
|that situation. |
| |
|Example: This square is formed using 4 toothpicks. |
| |
|A train of these toothpick squares is created horizontally like so: . . . |
| |
|Make a table and find a pattern that relates the number of squares in the train to the number of toothpicks needed to form that train. |
| |
| |
| |
| |
| |
| |
| |
|(e) Strategy: Try a Simpler Version of the Problem. You can begin to understand a complex problem by solving some scaled-down version of the |
|problem. Once you recognize a pattern in the way you are solving the simpler problems, then you can carry over this insight to attach the |
|full-blown problem. |
| |
|Apply Strategy to this Problem: Ten people are being honored for their work in reducing pollution. In how many ways can we line up these people |
|for a picture (all 10 in 1 row). Do not solve. Instead, state a simpler problem and solve it instead. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|(f) Strategy: Guessing is OK. One of the difficulties in solving word problems is that you can be afraid to say something that may be wrong and |
|consequently sit staring at a problem, writing nothing until you have the full-blown solution. Making guesses, even incorrect guesses, is not a |
|bad way to begin. It may give you some understanding of the problem. Once you make a guess, evaluate it to see how close you are to meeting all |
|the conditions of the problem. |
| |
|Apply Strategy to this Problem: Vince worked 18 hours last week. Part of the time he worked in a fast-food restaurant and part of the time he |
|worked tutoring a high school student in mathematics. He was paid $4.75 per hour in the restaurant and $15 per hour tutoring. If he earned |
|$126.50, how many hours did he work at each job? Make a guess, check to see how close it gets you to the correct answer, and then tell what that |
|guess tells you about the actual answer. |
| (g) Strategy: Convert a New Problem to an Older One. An effective technique in solving a new problem is to try to connect it with a problem|
|you have solved earlier. It is often possible to rewrite a condition so the problem becomes exactly like one you have seen before. |
| |
|Apply to Problem: Read Quiz Yourself 1 (p. 4 and full solution is on p.851) and compare it to example 8 in your textbook (p. 10). Your book |
|states that example 8 is “essentially the same problem as the four architects shaking hands in Quiz Yourself 1. Analyze what is alike in the two |
|problems that allows you to solve them the same way. Explain your thinking. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Fundamental Principles To Remember |
|For each of the following, look in your text and jot down several examples for each principle. |
| |
|The Always Principle: When we say a statement is true in mathematics, we are saying that statement is true 100 percent of the time. One of the |
|great strengths of mathematics is that we do not deal with statements that are “sometimes true” or “usually true”. |
| |
|For example: |
| |
| |
| |
| |
|The Counterexample Principle: An example that shows a mathematical statement fails to be true in a particular case is called a counterexample. |
|If you want to use a mathematical property and someone can find a counterexample, then the property you are trying to use is not allowable. |
|100 examples that a statement is true do not prove it to be always true. |
|1 example that a statement “fails” (doesn’t work), makes it a false statement. |
|Notice that when we say a statement is false, we are not saying that it is always false. We are only saying that the statement is not always true.|
|Since it is not always true, we cannot rely on it or use it to solve a problem. |
| |
|For example: |
| |
| |
| |
| |
| |
| |
| |
| |
|The Order Principle: Pay careful attention to the order in which operations must be performed. |
|Pay attention to when order makes a difference and when it does not. |
|Be sure you understand the conditions that allow you to change the order without changing the meaning. |
| |
|For example: |
| |
| |
| |
| |
| |
| |
|The Splitting Hairs Principle: Mathematics is much pickier than everyday language. Learn to “split hairs” when reading mathematical terminology. |
| |
|If two terms are similar but sound slightly different, they usually do not mean exactly the same thing. “Set A is equal to set B” does not mean |
|the same thing as “Set A is equivalent to set B”. |
|If two notations are similar, but slightly different, the same is true. They usually do not mean exactly the same thing. [pic]does not mean the |
|same thing as [pic]. |
|When two terms, notations, or ideas seem similar, but slightly different, you need to consciously work to get a clear idea of exactly what the |
|difference is. Not making the proper distinctions is often the cause of errors. |
| |
|For example: |
| |
| |
| |
| |
| |
| |
| |
|The Analogies Principle: Much of the formal terminology that we use in mathematics sounds like words that we use in everyday life. Whenever you |
|can associate ideas from real life with mathematical concepts, you will better understand the meaning behind the mathematics you are learning. |
|Always take the time to consciously connect new ideas to related ideas you already know. This is also a powerful memory aid. |
| |
|For example: |
| |
| |
|The Three-Way Principle: Explore mathematical ideas verbally, graphically, and by example. Three approaches gives you a deeper understanding of |
|an idea. |
|. |
|Verbally |
|Make analogies. |
|State the problem in your own words. |
|Compare it with other math situations you have experienced |
|Graphically |
|Pictures |
|Diagrams |
|Graphs |
|Tables |
|By Example: make numerical or other kinds of examples to illustrate the situation |
| |
|For example: |
| |
| |
|The Nature of Mathematics as an Academic Discipline (Section 1.3) |
| |
|Arithmetic is only a small part of the field of mathematics. |
|Mathematics depends on precise definitions and explicitly stated rules of logic. |
|Mathematical Truth is determined by the definitions and logic being applied. |
|The mathematical idea of a “set” helps us be precise and define what we are talking about. |
| |
|So the first two topics we study in this liberal arts survey of mathematics are |
|sets |
|logic |
| |
|Notation: the way we write things down in mathematics must reflect this precision. In this course you are required to use “good” notation in all|
|of your written work. |
|Introduction to Mathematical Sets |
| |
|Set: a set is a collection of objects. |
|Element or Elements of a set: any of the objects in the collection. These are also sometimes called members of the set. |
|( is the symbol that means “is an element of”. |
|( is the symbol that means “is not an element of”. |
| |
|“List” or “Roster” Notation: the set is defined by listing all of its elements between set braces { }. Sets are often |
|“named” by assigning capital letters to stand for the whole set. |
| |
| |
| |
|Set Builder Notation: the set is described by stating some characteristic that all the elements in the set have in common and is not satisfied by |
|any other object. Whether or not an object has this characteristic determines whether the object is in the set. |
| |
| |
| |
|Universal Set: the set U of all elements under consideration in a given discussion or problem is called the universal set. |
|Empty Set: The empty set has no elements. (If a set has any elements at all, it is not empty). The notation for “empty set” is either { } |
|or ( . If an equation has no solutions, then the solution set for the equation is the empty set. Another name for the empty set is a null |
|set. |
|Class Practice |
| |
|For [pic]all the past presidents of the United States}, is this “list” notation or “set builder notation”? |
| |
|(a) Fill in the blanks in the statements below with (or ( to make each statement true. |
|Bill Clinton _______ [pic] |
|Ariel Sharon _______[pic] |
|Hillary Clinton _______ [pic] |
|George H.W. Bush _______ [pic] |
|George W. Bush ________ [pic] |
|(b) Describe what you would consider to be the Universal set. |
| |
| |
| |
| |
| |
| |
|(c) Describe a related set that would be an empty set. |
| |
| |
| |
| |
| |
|(d) Critique the set-builder description used for this set. In what way might this description be deficient? How could we correct that? |
| |
| |
| |
| |
| |
| |
|Characteristics of Sets |
| |
|Some characteristics of sets: |
| |
|Order within the set: the order of the elements in the set does NOT matter. |
|Example: {1, 2, 3} is the same set as {2, 3, 1} |
| |
|Distinct elements: the elements of a set should be distinct, that is, they should each be different from the other. There should be no repeats |
|listed as part of the set. |
| |
|Cardinal number of a set: The cardinal number of a set tells the number of distinct elements within the set. It is important that we do NOT |
|repeat elements within the set so that when we count them we get the cardinal number of the set. The notation for “the cardinal number of set A”|
|is [pic]. |
| |
|Finite set: A set is finite if its cardinal number s a whole number. |
| |
|Infinite set: An infinite set is one that is not finite. |
|Example: T = {1, 2, 3, 4, . . . } means the set T contains the numbers 1, 2, 3, 4, and so on continuing in this pattern. The “ . . . ” means |
|“continuing the pattern”. When it is the last thing in the list, it means that you continue the pattern forever. Such as set is an infinite set |
|because the number of elements in the set is uncountable, they go on forever. |
| |
|Well-defined: A collection is well-defined if there is not ambiguity as to whether something belongs to the collection or not. So a set is |
|well-defined if we are able to tell whether or not any particular object is an element of the set. |
|Class Practice |
| |
|1. A = { 1, 2, 3, 4 }. Is {4, 3, 2, 1} the same set? Why or why not? |
| |
| |
|2. {(} is not an empty set. Why not? |
| |
| |
| |
| |
|3. V = {2, 4, 6, 8, . . . 26} the “ . . . ” is followed by the ending number. In this case we are told to continue the pattern up through 26 |
|and stop. This is a finite set because you can count the number of members in the set. What is [pic]? |
| |
|Common Number Sets |
|SPECIAL SETS OF NUMBERS: |
| |
|N = Natural numbers = {1, 2, 3, 4, . . . } (positive counting numbers) |
| |
|W = Whole numbers = {0, 1, 2, 3, . . . } (positive counting numbers and zero) |
| |
|J = Integers = { . . . (2, (1, 0, 1, 2, 3 . . . } (positive and negative counting numbers and zero) |
| |
|Q = Rational Numbers = [pic] |
|(numbers that can be written as fractions of integer numbers) |
| |
|R = Real Numbers: all the numbers that express distances from the origin on a number line. |
| |
|Real Numbers include: |
|all integers (positive counting numbers, negative counting numbers, and zero) |
|all fractions and decimals (non-integer, rational numbers) |
|irrational numbers such as [pic], e which cannot be written as fractions or as decimals that end or repeat. Irrational numbers are decimals |
|that never end and never repeat. This is why we frequently use a symbol, such as ( or e, to represent the number, because anything else we would |
|write would just be an approximation.. |
|Applying the definitions: Put a ( to indicate which set each number belongs to. |
|NOTE: a number may belong to more than one set. |
| |
|Number |
|N |
|W |
|J |
|Q |
|R |
| |
|7 |
| |
| |
| |
| |
| |
| |
|0 |
| |
| |
| |
| |
| |
| |
|(4 |
| |
| |
| |
| |
| |
| |
|2.4 |
| |
| |
| |
| |
| |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
|2( |
| |
| |
| |
| |
| |
| |
|.121221222… |
| |
| |
| |
| |
| |
| |
|.2323232323… |
| |
| |
| |
| |
| |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
| |
|Terminology Examples |
| |
| |
|1. List the set of integers between [pic]and 2, not inclusive. ____________________________ |
| |
| |
|2. List the set of multiples of 5 greater than 10. _____________________________________ |
| |
| |
|3. List the set of natural numbers between [pic]and 2, inclusive. _________________________ |
| |
| |
|4. Express {1, 2, 3, 4, 6, 12} as a set using set builder notation. |
| |
| |
|_____________________________________________________________________ |
| |
| |
|5. Express [pic] in roster notation. |
| |
| |
|_____________________________________________________________________ |
| |
|6. Express [pic]using the listing method. |
| |
| |
|_____________________________________________________________________ |
| |
| |
|7. Express [pic] in roster notation. _________________ |
| |
| |
|8. List the set of integers which when squared equal 11. _____________________________ |
| |
| |
| |
|9. List the whole number multiples of 2. ________________________________________ |
| |
| |
|10. For [pic] [pic]_______________ |
| |
|Comparing Sets (Section 1.4) |
| |
|Equal, Equivalent, or Not? |
| |
|In mathematics, language is used very precisely. The words “equal” and “equivalent” do not mean the same thing when applied to sets. |
| |
|equal sets: have exactly the same elements in them (not necessarily in the same order). |
| |
|equivalent sets: have exactly the same number of elements in them (not necessarily the same elements). |
| |
|Practice: Circle the correct term (equal/equivalent) to make each statement true. |
| |
|A ={ c, a, t} is equal/equivalent to B = {a, c, t} |
| |
|C = { 1, 2, 3, 4 } is equal/equivalent to D = {2, 4, 6, 8 } |
| |
|Subsets |
| |
|Subset: Set A is a subset of Set B if all the elements of set A are also contained in set set B. |
| |
|You can always tell if set A is a subset of set B by asking: “is every element of set A also an element of set B?” If every element in A is also |
|in B, then A is a subset of B. |
| |
|Practice: Circle the sets below that qualify as “subsets” of set G = {0, 1, 2, 3, 4, 5}. |
| |
|A = {0, 2, 4} B= {0, 1, 2, 3, . . . } C= {0, 1, 2, 3, 4, 5} D = {0, 1, -1, 2, -2} E = {1} |
| |
|According to the definition of subset, is a set a subset of itself? Why or why not? |
| |
| |
| |
| |
| |
| |
|is the symbol for subset. ( is the symbol for “is not a subset”. |
| |
|Fill in the blanks below to make true statements using ( or (. |
| |
|a. { 1, 2, 3} ______ W b. W ______ N c. {2, 4, 6} ____ {1, 2, 3, 5, 6} |
| |
|d. N _______ W e. W ______ J f. {4, 5} ______ { 4, 5} |
|Proper Subset |
| |
|Proper Subset: There is a special name and symbol for a set that is a subset and is not identical to the other set. Such a set is a Proper |
|Subset. ( is the symbol for proper subset. Notice there is no “or equal to” bar underneath it. A proper subset cannot be equal to the set it |
|is being related to. |
| |
|Examples: |
| |
|{1, 2} is both a subset and a proper subset of the set N. |
| |
|{1, 2} ( N and {1, 2} ( N |
| |
| |
|{ 1, 2, 3, 4, . . . } is a subset of N but is not a proper subset of N. |
| |
|{1, 2, 3, 4, . . . } ( N but {1, 2, 3, 4, . . . } ( N |
| |
|Practice: Circle the symbols that make the statement true. More than one symbol may apply. |
| |
|a. {1, 2} (, (, {1, 2, 3} b. W (, ( N c. i. {0} (, ( ( |
|(, ( (, [pic] (, ( |
| |
| |
|d. Is ( ( {1}????? Why or why not? |
| |
| |
| |
|e. Is ( ( N???? Why or why not? |
| |
| |
| |
|Five properties to remember about subsets: |
| |
|Every set is a subset of itself (() |
|The empty set is a subset of every set |
|For a set with n elements, there are [pic]distinct subsets. |
|For a set with n elements, there are [pic] (why would this be so?) |
|Pascal’s Triangle gives you the number of each-size subsets of a set |
| |
|Practice: Write out all of the subsets of the set A = {a, e, o} How many should there be? |
| |
| |
|Forming Pascal’s Triangle |
| |
|Pascal’s Triangle |
| |
|____ |
|____ ____ |
|____ ____ ____ |
|____ ___ ____ ____ |
|____ ____ ____ ____ ____ |
|____ ____ ____ ____ ____ ____ |
|____ ____ ____ ____ ____ ____ ____ |
|____ ____ ____ ____ ____ ____ ____ ____ |
|Pascal’s Triangle and Finding all the Subsets of a Set |
| |
|Use Pascal’s Triangle to be sure you find all the subsets of A = {a, b, c, d, e} |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Set Operations (Section 1.5) |
| |
|An operation in mathematics is a rule for getting an answer. In arithmetic, the four basic operations are addition, subtraction, multiplication, |
|and division. Notice that these familiar operations operate on numbers and not on sets. |
| |
|Set Operations operate on sets. The two basic set operations are Union ( and Intersection (. |
|Union of Sets |
|Union of two sets: The set of all the elements that are in either of the two sets being joined. |
|Example: For A = {4, 5} and B = {5, 6, 7}, then, A ( B = { 4, 5, 6, 7} |
|Notice that you do not write the 5 in the answer set twice. |
| |
|Venn Diagram: |
|Notice all of A is shaded |
|As well as all of B. |
| |
|Mathematical meaning of “OR”: All of the elements in A or in B or in both. |
|Intersection of Sets |
|Intersection of two sets: The set of only the elements that are in common between the two sets. Intersection is the OVERLAPPED part. |
|Example: For A = {4, 5} and B = {5, 6, 7}, then, A ( B = {5} |
|Venn Diagram: |
| |
| |
| |
| |
|Mathematical meaning of “AND”: All of the elements that are in both set A AND set B (at the same time). |
|Complement of a Set |
| |
|Complement of a set: The complement of any set is the set of all the elements in the given universal set, U, that are not in the set A. You |
|can think of “the complement of A” as “not A”. |
|The symbol for complement of a set is the prime ( . So A( means “the complement of set A”. |
| |
|In set builder notation [pic] |
| |
| |
| |
| |
| |
| |
| |
| |
|Difference of Two Sets |
| |
|The difference of sets B and A, written [pic], |
|is the set of elements that are in that are in B but not in A. |
| |
| |
|Shade [pic] in the Venn diagram at the right. |
| |
| |
|Write [pic] using set builder notation:__________________________________ |
| |
|Practice: Use U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Find these sets with respect to this U. |
| |
| |
|a. A = { 1, 2, 3}. A( = _______________ b. T = {0, 2, 4, 6, 8} T( = __________ |
| |
| |
|c. H = (. H( = ______________ d. U( = __________________________ |
| |
|Let [pic]= {0, 1} [pic]= { 1, 2, 3, 4 } [pic]= {2, 4, 6, 8 . . . } |
|Draw the Venn diagram for each of the following and then write the resulting set in list form. |
|a. [pic] b. [pic] c. [pic][pic] |
| |
| |
| |
|A Handy Relationship to Notice |
| |
|Complete the Venn Diagrams for each of the statements below and then compare the results: |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|[pic]:_______________________________________________ |
| |
|_________________________________________________________________________ |
| |
|_________________________________________________________________________ |
| |
| |
| |
|Now complete this description of the shaded region |
|of this Venn diagram: |
| |
|_Everything except ______________________________ |
| |
|_______________________________________________ |
| |
|Write three different set operation expressions that would result in the shaded region above. |
| |
| |
| |
| |
|De Morgan’s Laws for Set Operations |
| |
|[pic] [pic] [pic] |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Comparison:_______________________________________________________________ |
| |
|_________________________________________________________________________ |
| |
| |
|Complete the Venn Diagram for each of the statements below and then compare the results: |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
| |
| |
|Comparison:_______________________________________________________________ |
| |
|_________________________________________________________________________ |
|Survey Problems (Section 1.6) |
| |
|Survey problems (and similarly structured problems) are logic problems that are often found in real world situations. The logic of survey problems|
|can be used either to |
|Find missing values in data |
|Determine whether there is data missing or not |
| |
|They also appear on professional examinations like LSAT and MCAT and the GRE. They are considered an excellent way to evaluate a person’s ability |
|to think logically. |
| |
|Drawing a Venn diagram for a survey problem can take a complicated mess of words and numbers and turn it into an organized, manageable problem. |
| |
|Example: A survey was made of 200 city residents to study their use of mass transit facilities. According to the survey 76 used the train, 60 |
|used the subway, and 34 used the bus. 6 used the train and bus, 12 used the bus and subway, and 12 used the train and subway. 4 reported using |
|all three methods of mass transit. How many did not use mass transit at all? |
| |
| |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
|Determine which numbered regions make up the indicated set. |
| |
|1. [pic] _________________________________ |
| |
|2. [pic] _________________________________ |
| |
|3. [pic]_________________________________ |
| |
|4. [pic]_______________________________ |
| |
|5. [pic]____________________________ |
| |
|6. [pic]____________________________ |
| |
|Practice 1: In a certain group of 75 students, 16 students are taking psychology, geology, and English; 24 students are taking psychology and |
|geology; 30 students are taking psychology and English; and 22 students are taking geology and English. However, 7 students are only taking |
|psychology, 10 students are taking only geology, and 5 students are taking only English. |
|a. How many of these students are taking psychology? |
|b. How many of these students are taking psychology and English, but not geology |
|c. How many students in this group are not taking any of these three subjects? |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Practice 2: At a meeting of 50 car dealers, the following information was obtained; 12 dealers sold Buicks, 15 dealers sold Toyotas, 16 dealers |
|sold Pontiacs, 4 dealers sold both Buicks and Toyotas, 6 dealers sold both Toyotas and Pontiacs, 5 dealers sold both Buicks and Pontiacs, and one |
|dealer sold all three brands. |
|a. How many dealers sold Buicks and neither of the other two brands? |
|b. How many of the dealers at the meeting did not sell any of these brands of cars? |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Practice 3: In a recent survey of 300 people regarding television programming, the following information was gathered: 160 people watched ABC, |
|150 people watched CBS, and 150 people watched NBC, while 90 people watched both ABC and CBS, 70 people watched CBS and NBC, and 100 people watched|
|ABC and NBC. Forty people watched all three networks. |
|a. How many people watched ABC or NBC? |
|b. How many people watched only one of the networks? |
|c. How many people did not watch any of the networks? |
|How many people did not watch NBC? |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Practice 4: In a recent survey of parents of 160 third graders regarding the extracurricular activities their children are in, the following |
|information was gathered: |
| |
|28 children were in gymnastics |
|60 children were in dance |
|99 children were in swimming |
|13 children were in gymnastics and dance |
|21 children were in dance and swimming |
|19 children were in swimming and gymnastics |
|9 children were in all three activities |
| |
| |
| |
| |
| |
| |
|a. How many children were ONLY in dance and no other activity? |
| |
|b. How many children were in dance or swimming? (Remember what “or” means) |
| |
|c. How many children were in at least one of these activities? |
| |
|d. How many children were in at most two of these activities? |
|Problem 32: There are 95 students who have applied for a scholarship. If there are 41 men and 36 minorities, 19 of whom are women, how many women|
|applied for the scholarship? |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Problem 34: A survey is taken of 100 people who vacationed at a dude ranch. The following information was obtained: |
|19 took horseback riding lessons, attended the BBQ, and purchased a guide book. |
|34 attended the BBQ and purchased a guide book. |
|30 took horseback-riding lessons and purchased a guide book. |
|33 took horseback-riding lessons but did not attend the BBQ |
|86 took horseback-riding lessons or purchased a guide book |
|8 only purchased a guide book |
|3 did none of these three things |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|How many attended the BBQ or purchased the guide book? |
| |
|How many did not purchase the guide book? |
|Section 2.1 Inductive and Deductive Reasoning |
| |
|Inductive Reasoning is the process of drawing a general conclusion by observing a pattern of specific instances. This conclusion is called a |
|hypothesis or conjecture. |
| |
|1. Show how inductive reasoning can be used to determine the digit in the ones’ place (last digit) of this number without actually computing the |
|entire answer. |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Inductive reasoning is limited by the fact that no amount of examples that something is true will guarantee that it is ALWAYS true. We suspect it |
|is true but we really haven’t proved it. If at some point someone finds just 1 counterexample, then our conjecture (hypothesis) is proved to be |
|false. |
| |
| |
|Deductive Reasoning uses accepted facts and general principles to arrive at specific conclusions. It is a string of “if we know this, then we also|
|know this” type of reasoning. When we follow a set of rules to arrive at a conclusion, for instance in an algebra or geometry proof, or a |
|legal decision, we are using deductive reasoning. |
| |
|Legal example: the death sentence is only appropriate when the crime is of a particularly heinous and cruel nature. In this case, the defendant |
|tortured the victim prior to causing the victim’s death. Therefore the perpetrator of this crime deserves the death penalty. |
| |
|Deductive reasoning is powerful, but it is always dependent on the quality of the accepted facts and principles from which it is derived and the |
|skill with which those are used to reason. |
| |
|Commercial Example: This skin care product is made of all natural ingredients and is therefore good for you. The “accepted fact” is that if an |
|ingredient is “natural” it is automatically good for you. |
| |
|2. Explain how this “accepted fact” is generally used and then how it might lead to an incorrect conclusion in this case. |
| |
| |
| |
| |
| |
|Statements, Connectives, Quantifiers Section 2.2 |
| |
|Statement: in logic a statement is a declarative sentence that is either true or false. We represent statements by lower case letters such as p, |
|q, or r. |
| |
|Examples of things that are NOT statements: Questions, commands, exclamations, and paradoxes are not statements because they do not have the |
|quality of being “true” or “false”. |
| |
|Paradox: a sentence that cannot be assigned a truth value because it contradicts itself. Example: “This statement is false.” |
| |
|Truth value: The quality of being “true” or “false” is the statement’s truth value. |
| |
|Which of the following are statements in the mathematical sense? |
| |
|Statement (in mathematics) |
|Truth Value |
| |
| |
|February has 30 days |
| |
| |
| |
| |
|3 ( 2 = 4 + 2 |
| |
| |
| |
| |
|{1, 2, 3} = {4, 5, 6} |
| |
| |
| |
| |
|Stop the car! |
| |
| |
| |
| |
|All rules have exceptions |
| |
| |
| |
| |
|Types of Statements: |
| |
|Simple Statement: a statement that contains a single idea. Example: It is a sunny day. |
| |
|Compound Statement: a compound statement contains several ideas combined together. The words used to join the ideas of a compound statement |
|together are called connectives. |
| |
|5 Logical Connectives: |
|And (“but” is sometimes used in place of “and”) |
|as in “I am going, but Mary is not.” |
|Or |
| |
|Not |
| |
|If ( Then Regardless of order in everyday language, “IF” introduces the hypothesis and “THEN” introduces the conclusion. In symbols, the |
|hypothesis “IF”, ALWAYS goes first. |
|Example: If it rains, I will not go. I will not go if it rains. Both are written in symbols: |
| |
|If and Only If |
|Memorize these: |
| |
|Logical Connectives and their Special Mathematical Names and Symbols |
| |
|word or phrase |
|Special Math Name |
|Symbol |
| |
|and |
|(both, the overlap) |
|conjunction |
|( |
|(like intersection in sets) |
| |
|or |
|(one or the other, or both) |
|disjunction, |
|( |
|(like union in sets) |
| |
|if . . . then |
|(implies) |
|conditional |
|( |
| |
|If and only if |
|(iff) |
|biconditional |
|( |
| |
|not |
|negation |
|~ |
| |
| |
|In symbolic logic, we let letters stand for statements, the way we let letters stand for numbers in algebra. |
| |
|Translating the compound sentence: Today is Friday and I have a test. |
| |
|Let P = Today is Friday |
|Q = I have a test. |
| |
|Quiz Yourself #4-5 (p. 86, 87 of text) Write each statement in symbolic form: |
| |
|d: I will buy a DVD player. |
|i: I will buy an iPod. |
| |
| |
|(a) I will not buy a DVD player or I will not buy an iPod. __________________ |
| |
|(b) I will not buy a DVD player and I will buy an iPod. __________________ |
| |
|f: I fly to Houston |
|q: I will qualify for frequent flyer miles |
| |
|(a) If I do not fly to Houston, then I will not qualify for frequent flyer miles. |
| |
|____________________________ |
| |
|(b) I fly to Houston if and only if I will qualify for frequent flyer miles. |
| |
|____________________________ |
| |
|Negation applies only to the thing it is immediately next to. |
| |
|~P ( Q means _____________________________________________ |
| |
|~(P ( Q) means ____________________________________________ |
| |
|Clues for parentheses: |
| |
|“Ιt is false that” or “It is not true that” (everything after the word ‘that’ is in parentheses) |
| |
|Commas (either the phrase before the comma or the phrase after the comma will be in parentheses – which ever one is compound.) |
| |
|“Neither A nor B” means “not” ( A or B) |
| |
| |
|Important Counter-Intuitive Fact: (don’t let this trip you up) |
| |
|~ (A ( B) = ~A ( ~B (and is not equal to ~A ( ~B like you might think). |
| |
|Examples: Let P = Today is Monday |
|Q = Tomorrow is Wednesday |
|R = Tomorrow is Tuesday |
|T = Today is Tuesday |
| |
|It is false that today is Monday or tomorrow is Wednesday. |
| |
| |
| |
|Neither is today Monday nor is tomorrow Wednesday. |
| |
| |
| |
|If today is Monday, then tomorrow is Tuesday or tomorrow is Wednesday. |
| |
| |
| |
|Today is Tuesday if and only if tomorrow is Wednesday. |
| |
| |
| |
|Tomorrow is Wednesday implies that today is Monday. (we can write false statements) |
| |
| |
|Quantifiers state how many objects satisfy a given property. |
| |
|Universal Quantifiers state that all objects of a certain type satisfy a given property. |
|Examples: |
|“all” |
|“every” |
| |
|Existential Quantifiers state that there are one or more objects that satisfy a given property. |
|Examples: |
|“some” |
|“there exists” |
|“there is at least one” |
| |
|Negating Quantifiers |
| |
|Not “all are” has the same meaning as “At least one is not”. |
| |
|Not “some are” has the same meaning as “None are”. |
| |
|Venn Diagram Examples |
| |
|Statement “Everyday” Negation “Precise” Negation |
| |
|All Athletes are wealthy. Some athletes are not wealthy. At least one athlete is not wealthy |
|Not all athletes are wealthy. |
| |
| |
| |
| |
| |
| |
| |
|Statement “Everday” Negation “Precise” Form |
| |
| |
|Some students get a Some students do NOT No student gets a scholarship. |
|scholarship. get a scholarship. |
| |
| |
| |
| |
| |
| |
| |
|When asked to negate a statement in this class, use the “Precise” form. |
|More Practice with Quantifiers |
| |
|Use the precise, mathematical meanings when doing negations: |
| |
|“Negate ‘ALL’” has the precise, mathematical meaning “At least one is not”. |
| |
|“Negate ‘SOME ARE’” has the precise, mathematical meaning “None are”. |
| |
|(and vice-versa) |
| |
|Write in words, the negation of each of these statements: |
| |
|1. Some auto mechanics are incompetent. |
| |
|Negation: ____________________________________________________________ |
| |
|2. All married couples must file a joint tax return. |
| |
|Negation: ____________________________________________________________ |
| |
|3. All polygons have four sides. |
| |
|Negation: ____________________________________________________________ |
| |
|4. Some factories emit toxic wastes. |
| |
|Negation: ____________________________________________________________ |
| |
|Section 2.2 B Some Additional Details |
| |
|Dominance of Connectives |
| |
|What does this mean? |
|~(P(Q( ~R) (a negation) ???? |
|~P ( Q ( ~R ~P ( (Q ( ~R) (a conjunction) ????? |
|(~P (Q) ( ~R (a conditional) ????? |
| |
|Can’t leave this ambiguous. Either the parentheses are put in for you, or you need to follow the |
| |
|Order of Dominance: |
|Biconditional ( Compound statements on either side are considered in parentheses |
|Conditional ( Compound statements except those with ( are considered in parentheses |
|Conjunction (, Disjunction ( (Equal value, must indicate with parentheses) |
|Negation ~ (Only applies to the thing it is immediately next to) |
| |
|Practice: Add parentheses in each statement to form the type of compound statement indicated. |
|If none are needed, indicate that fact. |
| |
|Negation: ~ P ([pic]~Q |
| |
|b. Biconditional: P ( Q ( R |
| |
| |
|c. Disjunction: ~R ( Q ( P ( S |
| |
| |
|d. Conditional: ~R ( Q ( P ( S |
| |
| |
|e. Conjunction: ~R ( Q ( P ( S |
| |
| |
|f. Negation: ~R ( Q ( P ( S |
| |
| |
| |
| |
|Designating Definers |
| |
|For this section, first circle the dominant connective. |
| |
|Then define the phrases. Do not include the negative in the defined phrase. |
| |
|Then rewrite the entire statement in symbolic form, using the negation sign when appropriate. |
|Do the homework on worksheet 2.2B this way. |
| |
|Example: If the Vikings do not win the championship, then I won’t win the $500. |
| |
|p = __________________________________________________ |
| |
|q = __________________________________________________ |
| |
|Symbolic Statement: ___________________________________ |
| |
|Example: I will go to Minneapolis if and only if I have no homework over the weekend. |
| |
|p = __________________________________________________ |
| |
|q = __________________________________________________ |
| |
|Symbolic Statement: ___________________________________ |
|Relationships between Biconditional and Conjunction |
| |
|The biconditional, “if and only if” is like when we say “and vice-versa” in everyday language. |
| |
|It is the same as the conjunction of two conditionals, one the reverse order of the other. |
| |
|Example: I will go to Minneapolis if and only if I have no homework over the weekend. |
| |
|First conditional statement = ________________________________________________ |
| |
|Reverse conditional statement = _____________________________________________ |
| |
|Conjunction that is equivalent to the original biconditional statement: |
| |
|_______________________________________________________________________________ |
| |
|_______________________________________________________________________________ |
| |
|Equivalent Conjunction as a symbolic statement: _______________________________ |
| |
|Section 2.3 A Standard Set-Ups for Truth Tables |
| |
|All Possible Truth Values for a given number of statements: |
| |
|1 Statement Set-Up: |
| |
|P |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|2 Statements Set-Up: |
| |
|P |
|Q |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|3 Statements Set-Up: |
| |
| |
|P |
|Q |
|R |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Section 2.3 – 2.4 Making Sense of the Truth Tables for the Connectives |
| |
|Definers: |
|P= shape is a square |
|Q = color is blue |
| |
|Negation Conjuction |
| |
|P |
|~P |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Disjunction Conditional |
| |
| |
|Q |
|P( Q |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Biconditional |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Summary of Logic Rules |
| |
|Definers for Example: P = my shape is a square. Q = the color of my shape is blue. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Practice |
| |
|Complete the Truth Table and circle the final column (answer). |
| |
|P |
|Q |
|~ [pic] |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Complete the Truth Table and circle the final column (answer). |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Complete the Truth Table and circle the final column (answer). |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|“Logically Equivalent” |
| |
|Logically Equivalent: Two statements are logically equivalent if they have the same variables and, when their truth tables are compute3d, the |
|final columns in the tables are identical. |
| |
|De Morgan’s Laws for Logic: If p and q are statements, then: |
| |
|a) [pic] |
| |
|b) [pic] |
| |
|Show part (b) of DeMorgan’s Law for Logic is true, using the definition of “logically equivalent”. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Section 2.4B The Conditional and Related Forms |
| |
|The conditional is composed of two parts, the hypothesis (IF) and the conclusion (THEN). |
| |
|Converse: the converse of a conditional has the form [pic] |
| |
|Inverse: the inverse of a conditional has the form [pic] |
| |
|Contrapositive: the contrapositive of a conditional has the form [pic] |
| |
|ONLY THE CONTRAPOSITIVE is logically equivalent to the original conditional. |
| |
|Write in words, the converse, inverse, and contrapositive of the statement: |
| |
|Definers: m stands for “Marijuana is legalized”. |
| |
|d stands for “Drug abuse will increase”. |
| |
|Original: [pic]. In words: ____________________________________________ |
| |
|_____________________________________________________________________ |
| |
|Converse: ___________. In words: _______________________________________ |
| |
|______________________________________________________________________ |
| |
|Inverse: ____________. In words: ________________________________________ |
| |
|______________________________________________________________________ |
| |
|Contrapositive: _____________. In words: ________________________________ |
| |
|______________________________________________________________________ |
| |
|Show that the contrapositive of a statement is logically equivalent to the original statement. |
| |
| |
| |
| |
| |
| |
| |
| |
|Necessary and Sufficient |
| |
|Necessary and Sufficient Conditions |
| |
| |
|Venn Diagram Interpretation: |
| |
|Being in the Universe is necessary to being in set A. |
| |
|Being in Set A is sufficient to guarantee that you are in the Universe. |
| |
|[pic] |
| |
| |
| |
| |
|Which is how your book comes up with the rule: |
| |
|The hypotheses is “sufficient” and the conclusion is “necessary”. |
| |
|Other Clues to Translating Conditional Statements From Words to Symbols |
| |
|All of the following are the same as “if [pic] then [pic]”. Notice that some follow the order of the words and some reverse the order. |
| |
|Follows Order |
|Reverses Order |
| |
|If p, then q |
| |
| |
|p only if q |
|q if p |
| |
|p is sufficient for q |
|q is necessary for p |
| |
|To do p it is necessary that . . . q |
|To do q it is sufficient that . . . p. |
| |
| |
| |
|Use the definers: f represents “living in Fargo (of Cass County), |
|n represents “living in North Dakota” |
|c represents “living in Cass County (of ND). |
| |
|1. One can live in Cass County only if one lives in North Dakota. |
| |
| |
| |
|2. One is living in North Dakota if one is living in Fargo. |
| |
| |
| |
|3. “Living in North Dakota is not sufficient for living in Fargo. |
|Rule: All of the following are the same as “if [pic] then [pic]” |
| |
| |
| |
| |
| |
| |
| |
| |
|Then translate following the rules above. Then rewrite your translation in English words, in the form “If . . . Then” to see if it means the same|
|as the original sentence. |
| |
|___________4. If you give us your credit card number, then we will hold your reservation. |
| |
|____________________________________________________________ |
| |
|___________5. To hold your reservation, it is necessary for you to give us your credit card number. |
| |
|_________________________________________________________________ |
| |
|___________6. Your drivers’ license will be suspended if you are convicted of driving under the influence of alcohol. |
| |
|________________________________________________________________ |
| |
|___________7. To hold your reservation, it is sufficient to give us your credit card number. |
| |
|________________________________________________________________ |
| |
|____________8. You will graduate only if you have 2.5 grade average. |
| |
|________________________________________________________________ |
| |
|____________9. To qualify for a discount on your airline tickets, it is necessary to pay for them two weeks in advance. |
| |
|_________________________________________________________________ |
| |
|____________10. You can return the video game only if you have not opened the package. |
| |
|_______________________________________________________________ |
| |
|____________11. To graduate this semester, it is necessary that you complete 18 credits. |
| |
|_______________________________________________________________ |
| |
|____________12. To reserve a campsite, it is sufficient that you pay a small deposit. |
| |
|Section 2.5 Verifying Arguments |
| |
|A logical argument is a collection of statements that taken together produce a logical conclusion. |
| |
|The statements used in a logical argument are called premises. |
| |
|An argument is not called “true” or “false”. An argument is either “valid” or “invalid”. |
| |
|An argument is valid whenever all of the premises are true, then the conclusion must also be true. |
| |
|So we can make a truth table of a logical argument by connecting all the premises by the connective “and” and then checking the conditional – |
|whether it really “implies” the conclusion or not. |
| |
|One way that logical arguments are written is as follows: |
| |
|If Mark goes to West Acres Mall, he will buy a new video game. |
|Mark is going to West Acres Mall |
|------------------------------------------------- |
|[pic] Mark will buy a new video game. |
| |
|Symbolically, let |
|m represent the statement “Mark goes to West Acres Mall” and |
|g represent the statement “Mark will buy a new video game”. |
| |
|Then the argument can be written as: |
| |
|_________________________________ |
| |
|__________________________________ |
|----------------------------------------------------- |
| |
|_________________________________ |
| |
|We can verify validity of the argument by computing the following truth table: |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Valid or Invalid?????? |
|VALID Arguments |
| |
|These forms that are always VALID: |
| |
|Law of Detachment Law of Contraposition Law of Syllogism Disjunctive Syllogism |
| |
| |
| |
| |
| |
| |
| |
| |
|Show that the Law of Syllogism is a valid argument: |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|INVALID Arguments |
| |
|BEWARE of these argument forms. They are ALWAYS INVALID. |
| |
|Fallacy of the Converse Fallacy of the Inverse |
|(Fallacy of Affirming the Consequence) |
| |
| |
| |
| |
| |
| |
| |
| |
|Show that the Fallacy of the Inverse |
|is an Invalid Argument. |
| |
| |
| |
| |
|Identifying Common Valid and Invalid Argument Forms |
| |
|For each of the following arguments, |
|(a) underline the basic statements and assign each one a variable |
|(b) rewrite the statement in symbolic form |
|(c) identify the form of the argument and |
|(d) state whether the argument is valid or invalid. |
| |
|p. 120 #12 If news on inflation is good, then stock prices will increase. |
|News on inflation is good. |
|------------------------------------------------------------------------ |
|Therefore, stock prices will increase. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|p. 121 #18 If you perform maintenance on your PC, then you violate your warranty. |
|You do not perform maintenance on your PC |
|-------------------------------------------------------------------------------------- |
|Therefore, you do not violate your warranty. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|p. 121 #26 If you cut the amount of fat in your diet, then you will have more energy. |
|You don’t have more energy. |
|----------------------------------------------------------------------------------------- |
|Therefore, you did not cut the amount of fat in your diet. |
| |
| |
| |
| |
| |
| |
| |
|Section 2.5 Additional Material |
| |
|Formal Proofs using the Common Logical Forms as “Reasons”. |
| |
|Argument: |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Argument: |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Section 2.6 Using Euler Diagrams to Verify Syllogisms |
| |
|A syllogism is valid if whenever its premises are all true, then the conclusion is also true. |
| |
|If the conclusion of a syllogism can be false, even though all the premises are true, then the syllogism is invalid. |
| |
|Euler Diagrams (Venn Diagrams) can help us understand the conditions involved in an argument, and when we can show a counterexample or can picture |
|all possible cases, Euler Diagrams can be used to prove whether an argument is valid or invalid. |
| |
|Important Principles to remember about logical proofs. |
| |
|Counterexample: One counterexample proves the argument is invalid. |
| |
|Single Example of a Case that is Valid. One example that the argument is valid in a single case, proves nothing about the argument in general. It|
|is just one example. |
| |
|Checking all possible Cases. If you can determine that an argument must satisfy one of a finite number of cases (in this section, Euler Diagram |
|representations) and you can show that each and every one of those leads to the conclusion that the argument is valid, then that proves the |
|argument is valid. |
| |
| |
|Example 2 (p. 124) |
| |
|[pic] |
| |
| |
| |
| |
|Example 3 (p. 125) |
| |
|[pic] |
| |
| |
| |
|Example 4 (p. 126) |
| |
|[pic] |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Sections 12.1 - 12.2 |
|Counting Methods & The Fundamental Counting Principle |
| |
|Counting Method #1: Organized Lists: The most basic method of counting is by organized lists. |
| |
|We have already used this technique previously, when we listed all the possible subsets of a set. |
| |
|Example: List all the subsets of {A, B, C} |
|Use the number of elements in each subset as the organizing feature. |
| |
|0-element subsets 1-element subsets 2-element subsets 3-element subsets |
| |
| |
| |
| |
| |
| |
|Example: List all the possible birth orders of boys and girls in a family of 3 children. |
|Use the number of girls in each family of 3 children as the organizing feature. |
| |
|0-girls 1-girl 2-girls 3-girls |
| |
| |
| |
| |
| |
| |
| |
|Example: List all possible cards that could be chosen |
|when 1 card is drawn from a poker deck. |
|Use the suits as the organizing feature. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Counting Method #2: Tree Diagrams |
| |
|Example: How many possible outcomes when you toss a coin 3 times |
|and record heads or tails each time? |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Example: How many different outcomes when you toss a standard die and then flip a coin? |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Example: How many possible different budget lunches at the a buffet |
|where you choose 1 “vegetable”, 1 “potato”, and 1 “meat”? |
|choices for “vegetable” are peas, carrots, green beans, broccoli |
|choices for “potato” are mashed or baked |
|choices for “meat” are chicken, beef, fish |
|choices for “desert” are cake or ice cream |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Counting Method 3: The Fundamental Counting Principle (A short cut of the tree diagram) |
|To do a series of tasks, one after the other, when there are “a” ways to do the first task, “b” ways to do the second task, “c” ways to do the |
|third task, etc., there will be a total of [pic] ways to do the whole series of tasks. |
| |
|This is equivalent to finding the number of final branches in a tree diagram by multiplying the number of branchings at each stage. |
| |
|Example: How many different license plates can be formed using 6 symbols if the first three must be letters and the last three must be digits and |
|repeats are allowed? |
| |
| |
| |
| |
| |
|Example: How many ways to choose a 4-digit PIN if the first digit must be a 1 or a 5 and all the other positions could be any digit? |
| |
| |
| |
| |
| |
|Example: How many different 4-letter “code words” can be created using the letters T, A, R, E, P if a “code word” is defined to be any ordered |
|series of 4 of these letters with NO REPEATS. |
| |
| |
| |
| |
| |
|Example: How many even 4-digit numbers can be formed from these digits: 1, 2, 3, 4, 5 if repeats are not allowed. |
| |
| |
| |
| |
| |
|Example: How many ways can 4 people (Abby, Ben, Claire, and Denny) be seated in a row with four seats if Abby and Claire must sit next to each |
|other? |
| |
| |
| |
| |
| |
| |
| |
|Section 12.3 Permutations and Combinations |
| |
|Factorial: an arithmetic operation. The symbol for “factorial” is the exclamation point (!). |
| |
|5! = ___________________________ |
| |
|0! = ___________________________ |
| |
|6! = ___________________________ |
| |
|Combination: a grouping in which the order does NOT matter. |
| |
|Common Notation: [pic] means “n things taken in combinations of r at a time” |
|C(n,r) is the notation that your book uses. |
| |
|Example: From 5 colors, choose any three, any order. List all possible combinations. |
|Red, Green, Blue, Yellow, Purple |
| |
| |
| |
| |
| |
|Combination Formula: [pic] |
| |
|so [pic] |
| |
| |
|Note: If the order matters, then our list of 10 ways becomes much larger: |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|The preceding list of ordered groups is the list of all the permutations of n things taken r at a time. |
| |
|Permutation: a grouping in which the order DOES matter. |
| |
|Common Notation: [pic] means “n things taken in permutations of r at a time”. |
|P(n,r) is the notation your book uses. |
| |
| |
|Combination Formula: [pic] |
| |
|so [pic] |
| |
| |
|Clues about which formula to use when: |
| |
|Combinations |
|Permutations |
| |
| |
|“in any order” |
| |
|Choose a committee, equal ranks |
|(all members are equal) |
| |
|Choose a team, equal ranks |
|(all players are equal) |
| |
|Choose who can go in the car |
| |
|Choose who can sit at the table |
| |
|“how many ways to order” |
| |
|Choose a committee with ranks like chairperson, secretary, treasurer, etc. |
| |
|Choose a team with ranks like playing positions, batting orders, captain, etc. |
| |
|Choose the seating within the car |
| |
|Choose seating at the table |
| |
| |
| |
|Section 13.1 Probability: What are the Chances? |
| |
|Probability studies were a result of mathematicians studying gambling problems. The Mathematician Blaise Pascal (1623 –1665) solved two gambling |
|problems posed by a professional gambler and began several correspondences with other mathematicians on the subject of probability. |
| |
|Probability is the mathematics of chance. It describes the predictable long-run patterns of random outcomes. For instance, if you toss a fair coin|
|a single time, the outcome (heads or tails) is completely random and unpredictable. But if a coin is tossed 10,000 times, we can be sure that it |
|will come up heads about half the time. |
| |
|Probability is expressed as the relative frequency with which we can expect an event to occur. This relative frequency is a ratio and can be |
|expressed as a fraction, a decimal, or a percent. |
| |
|Experiment: an activity where the results can be observed and recorded as a measurement. |
| |
|Outcome: what happens when the experiment is performed. Each of the possible results of an experiment is an outcome. |
| |
|Frequency: how often a particular outcome occurs. |
| |
|Sample Space is the set of all possible outcomes for an experiment. |
| |
|Event: any subset of the sample space. |
| |
|Target event: the event we are interested in finding the probability for. |
| |
|Probability of an outcome in a sample space: is always a number between 0 and 1 inclusive. |
| |
|The sum of the probabilities of all of the outcomes in the sample space must be 1. |
| |
|The probability of an event E, written P(E), is defined as the sum of the probabilities of the outcomes that make up event E. |
| |
| |
|Empirical (Experimental) Probability |
| |
|Empirical (also called “Experimental”) Probability: We actually do the experiment and use the actual data to compute the probability. |
| |
|P(event) =[pic] |
|Five things to know about experimental probabilities |
| |
|Experimental probabilities come from doing the actual experiments. |
|When an experiment is done several times, each time is called a trial. |
|Experimental probabilities may vary slightly from trial to trial. |
|Experimental probabilities are approximations of theoretical probabilities. |
|Law of Large Numbers: as the number of trials gets very big, the experimental probability gets closer and closer to the theoretical probability. |
|Theoretical Probability |
| |
|Equally likely Outcomes: when each outcome of an experiment is as likely to occur as any other outcome. |
| |
|Theoretical Probability is probability is computed using the laws of probability rather than doing the experiment. It is based on the concept of |
|“equally likely outcomes”. |
| |
|[pic]= [pic]= set of equally likely outcomes of an experiment |
| |
|[pic]the number of outcomes (equally likely) in the Sample Space |
| |
|[pic]subset of the (equally likely) outcomes from the Sample Space that form the “Target” |
| |
|[pic]the number of equally likely ways the target event can happen |
| |
| |
|[pic]event) = [pic] = [pic] |
|Example |
| |
|During WWII, John Kerrich, while a prisoner of war, tossed a coin 10,000. He got 5067 heads. |
| |
|What is the experiment? |
| |
|List the Sample Space: |
| |
|What is the Target Event? |
| |
|Write this experimental probability as a fraction, as a decimal, and as a percent. The fraction will be exact, but often the decimal and percent |
|must be rounded off. |
| |
|Fraction Rounded Decimal Rounded Percent |
|P(H) = ( ( |
| |
| |
|Is the empirical or theoretical probability? |
| |
| |
|What is the theoretical probability for this experiment? |
| |
| |
|Basic Properties of Probability |
| |
|1. [pic] |
| |
|2. [pic] |
| |
|3. P (S) = 1 |
|Example: |
| |
|Experiment: spin the spinner (what makes this an equally likely situation?) |
| |
|[pic] |
| |
| |
|Compound Events – 2 ways to find sample spaces for compound events |
| |
|Compound Event: An outcome in an experiment that consists of more than one event. For instance, tossing 2 die, one red and the other white. The |
|outcomes are recorded in the order that they occur, usually in ordered n-tuples like (red #, white#). |
| |
|Cartesian Products: When there are exactly two parts to the experiment, you can generate the entire sample space using a Cartesian product table: |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Tree Diagrams: When there are two or more parts to the experiment you can use a tree diagram to generate the entire sample space. Example: toss |
|a die, flip a coin, spin this spinner. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|The theoretical probability of a compound event is the product of the probabilities of the component events. If you fill in the probability of |
|each branch along the way, the probability of the final outcome is the product of the probabilities along the branches. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Book Example: Probability theory used to explains Genetics (pp. 728-720) |
| |
|Gregor Mendel (1800’s) studied genetics by cross-breeding pea plants He started with pure strains of plants with particular observable |
|characteristics: |
| |
|1. tall vs. short; |
|2. yellow seed vs. green seed; |
|3. smooth seeds vs. wrinkled seeds |
| |
|From his data he concluded that each plant had dominant and recessive characteristics that combined. For instance, when he cross-bred yellow seed |
|plants with green seed plants he found that the Yellow (Y) characteristic was dominant and the green (g) characteristic was recessive. |
| |
|His results: in the first generation of cross breeding yellow-seed plants with green-seed plants, in the second generation his plants produced |
|6,022 plants with yellow seeds and 2,001 plants with green seeds. Notice that means that roughly ¾ of the plants in the second generation had |
|yellow seeds while ¼ of the plants in the second generation had yellow seeds. |
| |
|How did I get those fractions? |
| |
| |
| |
| |
|His explanation related to probability. Consider the ways the characteristics could combine: |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Odds |
| |
|Odds are reported as ratios either “in favor” or “against” a target event happening. |
|If the outcomes are equally likely, and there are 7 ways to win (for the target event to happen) and 3 ways to lose (the target event doesn’t |
|happen), then the |
|“Odds in favor” of the event are 7:3 or [pic] |
|“Odds against” the event are 3:7 or [pic] |
| |
|CAUTION: ratios, even when written in [pic] form, are not exactly the same as fractions! |
| |
|Remember that ratios relate “parts” to “parts” |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Other Probability Problems |
| |
|1. If we toss four fair coins, what is the probability that we get exactly two heads? |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|2. Two cards are selected without replacement from a set of five cards having a picture of a star, a circle, wiggly lines, a dollar sign, and a |
|heart. What is the probability that no star appears on either card? |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|3. A 5-card hand is drawn at random from a 52-card deck. What is the probability that all the cards are face cards? |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|4. Experiment: Spin this spinner and record the color the pointer lands on. |
| |
|a. List the sample space:_______________________________ |
| |
|b. P(B) = _________ |
| |
|c. P(R) = _________ |
| |
|d. P(Y or R) = __________ |
| |
|e. P(G and B) = __________ |
| |
|f. P(G or B) = _________ |
| |
|g. P(R or Y or B) = __________ |
| |
|5. Experiment: A bag has 13 red marbles, 3 blue marbles, 7 green marbles, and 2 yellow marbles. Draw 1 marble and record the color. |
| |
|a. List the sample space: _______________________________________________ |
| |
|b. P(blue or green or yellow) = _____________ |
| |
|c. P( orange and yellow) = _____________ |
| |
|d. P(red or blue or green or yellow) = ______________ |
| |
|e. P(blue and red) = ______________ |
| |
| |
| |
| |
| |
|6. Experiment: Toss a coin 3 times and record the number of heads that occur within the 3 tosses. |
| |
|a. List the sample space: _________________________________________________ |
| |
|b. P(2) = _______________ |
| |
|c. P(0) = _______________ |
| |
|d. P(1 or 2 or 3) = __________________ |
| |
|7. Experiment: Toss a coin 5 times and record whether each toss is heads or tails. |
| |
|a. describe the sample space: |
| |
| |
|b. how many items will be in the sample space? |
| |
| |
| |
| |
|8. Experiment: Toss a coin 4 times and record whether it lands heads or tails on each toss. |
| |
|a. Show how to compute the number of items in the sample space. |
| |
| |
|b. List the sample space in an orderly list. |
| |
| |
| |
| |
| |
| |
| |
| |
|c. P(HTHT) = ____________ |
| |
|d. P(HTHTH) = ___________ |
| |
|e. P(exactly 2 heads) = _____________ |
| |
|f. P(at least 1 tail) = _______________ |
| |
|g. P(not more than 2 heads) = _____________ |
| |
| |
\
|Section 13.2 Complements and Unions of Events |
| |
|Target Event: the set of outcomes that are considered “wins”. |
| |
|Complement of a Target Event: the set of all outcomes that are NOT in the Target Event. |
| |
|Example: Target Event A = {King of Hearts, King of Diamonds, King of Spades, King of Clubs} |
| |
|Then the complement of A, [pic]= __________________________________________ |
| |
|P(A) = __________ P([pic]) = ___________ |
| |
|Rule: If E is an event, then [pic] |
| |
|Practice: If the probability of candidate B winning the election is calculated to be .42, what is the probability that candidate B does NOT win |
|the election? |
| |
| |
|Mathematical “or” means “1 or the other or both” and is found with the set operation Union. |
| |
|Be careful with computing probabilities of sets connected with “or”. Some sets have no overlap and you can simply add the probabilities. But sets|
|that have overlap, you must adjust your answer so that you do not count the overlap twice. |
| |
|Example: Target Event C = {any card that is a Jack or a club} |
| |
|[pic] |
| |
|[pic] |
| |
|[pic] |
| |
|Rule for Computing the Probability of a Union of Two Events: |
| |
|[pic] |
| |
|Mutually Exclusive: Events E and F are mutually exclusive if [pic]is the empty set. |
| |
|Practice: [pic] |
| |
|[pic] |
| |
|[pic] |
|Practice: Joanna earns both a salary and a monthly commission as a sales representative for an electronics store. The following table lists her |
|estimates of the probabilities of earning various commissions next month. |
| |
|(a) P(she will earn $1,500 in commissions) |
| |
| |
| |
|(b) P(she will earn less than $2,250 in commissions) |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Practice: (p. 744 problem #22) If we draw a card from a standard 52-card deck, what is the probability that the card is neither red nor a queen?|
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|Section 13.3 Conditional Probability and Intersections of Events |
| |
| |
| |
|Conditional Probability: P (F | E) means the conditional probability of F, given that E has already occurred. Notice that the order within the |
|parentheses makes a difference in what the problem is asking and what the answer will be. |
| |
| |
| |
|Counting Rule: [pic] whenever E and F are events in a sample space with |
|equally likely outcomes. |
| |
| |
| |
| |
|Probability Rule: [pic] |
| |
|[pic] |
| |
|Example: If we select a graduate who was offered between $30,001 and $35,000, what is the probability that the student has a degree in the health |
|fields? |
| |
|Which event is event “E”, the one that has already happened? E = ______________ |
| |
|Which event is event “F”, the one we are going to find the probability for |
|after event E has already happened? F= ______________ |
| |
|P(F|E)= |
| |
| |
| |
| |
|[pic] |
| |
| |
|Example: If we select a graduate who received more than $35,000 as a starting salary, what is the probability that the graduate has a degree in |
|technology? |
| |
|Which event is event “E”, the one that has already happened? E = ______________ |
| |
|Which event is event “F”, the one we are going to find the probability for |
|after event E has already happened? F= ______________ |
| |
|P(F|E)= |
| |
| |
| |
| |
|Example: Suppose that we draw two cards without replacement from a standard 52-card deck. What is the probability that both cards are kings? |
| |
|Notice that we can compute the probability that the first card is a king, but unless we KNOW what card gets picked first, we cannot compute the |
|probability that the second card is a king: |
| |
| |
| |
| |
| |
|(a) P(second card drawn is a King) = ______ |
| |
|(b) P(both are Kings) =_______ |
| |
| |
| |
| |
| |
| |
|Example: A person applying for a driver’s license first must pass a preliminary written test. After passing that test, the person must take the |
|driving test. If a candidate for a license fails the driving test 3 times, the person must go back and retake the written test before being |
|allowed to take a driving test again. |
| |
|Suppose 60% pass the driving test the first time, 75% pass on their second try, and only 30% pass on their third try. |
| |
|(a) Draw a tree diagram showing this situation |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|(b) What is the probability that a candidate will first fail the first time and then pass the test on the second try? |
| |
| |
| |
|(c) What is the probability that a candidate will fail twice and pass the test on the third try? |
| |
| |
| |
| |
| |
|Independent and Dependent Events |
| |
|Independent events have no effect on each other’s probabilities. |
| |
|Definitions: |
| |
|Independent Events: Events E and F are independent events if [pic]. |
| |
|Dependent Events: When [pic] events E and F are Dependent Events. |
| |
|Example: F = “a five shows on a red die” |
| |
|G= ”the total showing on the dice is greater than ten” are independent events or not. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|(a) Determine whether or not the events F and G are independent using the definition of independence. |
| |
| |
|(b) Explain in your own words what this means. |
| |
| |
|Example: In a sample space S, [pic] |
| |
|(a) draw a probability Venn diagram for this situation |
| |
| |
| |
| |
| |
| |
|(b) Compute [pic] (c) Compute [pic] |
| |
| |
|(d) Compute [pic] (e) Compute [pic] |
| |
-----------------------
U
A
If set A is represented by the circle and the Universe is represented by the rectangle, then A(, (the complement of A or sometimes called “not A”) is represented by the shaded region.
B
A
A
B
B
A
U
C
B
U
B
A
U
A
C
B
A
C
B
B
A
U
C
A
A
C
B
A
C
B
A
C
B
U
U
B
A
U
B
A
U
B
A
U
B
A
B
A
W
T
S
B
Y
X
U
r1
r2
r3
r4
r5
r6
r7
r8
G
In symbols:
c
Y
B
s
c
r
|Follows Order |Reverses Order |
|If p, then q | |
|p only if q |q if p |
|p is sufficient for q |q is necessary for p |
|To do p it is necessary that . . . q |To do q it is sufficient that . . . p. |
c
r
“sufficient”
“necessary”
A
U
|[pic] |[pic] | |
| | | |
| | | |
| | | |
| | | |
|[pic] |[pic] |[pic] | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
|P |Q |P [pic] Q |
| | | |
| | | |
| | | |
| | | |
a
O
Y
B
O
B
r
|P |Q |R |[pic] |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
|P |Q |~ [pic] |
| | | |
| | | |
| | | |
| | | |
“If you pass this class, I will take you to lunch.”
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Note that if the results in the final column are all true, this guarantees that all the premises must be true, since the conjunction of statements is only true when all the premises are true.
|[pic] |[pic] |[[pic] |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
|P |Q |P ( Q |
| | | |
| | | |
| | | |
| | | |
|P |Q |P(Q |
| | | |
| | | |
| | | |
| | | |
B
Biconditional is
true only when both statements have the same truth value.
[both true or both false]
|P |Q |P ( Q |
|T |T |T |
|T |F |F |
|F |T |F |
Conditional is
false only when first statement (antecedent) is true and the second statement (consequent) is false.
[False only when T leads to F]
|P |Q |P ( Q |
|T |T |T |
|T |F |F |
|F |T |T |
|F |F |T |
Disjunction is
only false when both statements are false.
|P |Q |P ( Q |
|T |T |T |
|T |F |T |
|F |T |T |
|F |F |F |
Conjunction is
true only when both statements are true.
|P |Q |P(Q |
|T |T |T |
|T |F |F |
|F |T |F |
|F |F |F |
Negation reverses the truth values.
|P |~P |
|T |F |
|F |T |
Y
B
O
B
Y
B
O
B
a
d
r
a
o
G
c
r
d
(b)
(c) _____________________________
(d) _____________________________
(b)
(c) _____________________________
(d) _____________________________
(b)
(c) _____________________________
(d) _____________________________
Statement Reason
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
Statement Reason
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. [pic]
10. [pic]
11. [pic]
Red Blue
Yellow Red
| |1st generation Plant #2 |
| | |Y |g |
|1st generation | | | |
|Plant #1 | | | |
| | | | |
| |Y | | |
| | | | |
| |g | | |
Example: Cartesian product of two sets.
How many possible outcomes if you toss
1 red and 1 white die?
The ratio of shaded squares to non-shaded squares is 3:1 or [pic]
The fraction of shaded squares is [pic].
In a fraction, the bottom number must tell the number of equal pieces in the whole.
In a ratio, the bottom number is the part being related to and does not have to be the number of the pieces in the whole.
Example of Cartesian Product
for experiment: tossing two dice
| |1 |2 |3 |4 |5 |6 |
|1 | | | | | | |
|2 | | | | | | |
|3 | | | | | | |
|4 | | | | | | |
|5 | | | | | | |
|6 | | | | | | |
Example of Cartesian Product
for experiment: toss a die
and spin this spinner
| |R |B |Y |G |
|1 | | | | |
|2 | | | | |
|3 | | | | |
|4 | | | | |
|5 | | | | |
|6 | | | | |
Red
Blue
Yellow Green
red
blue
yellow
Your book calls this a “Punnett square” when it is used for figuring genetic outcomes. There are two in your homework assignment.
(a) P(second card drawn is a King) = ______
(b) P(both are Kings) =_______
(1,1) (2,1) (3,1) (4,1) (5,1) (6,1)
(1,2) (2,2) (3,2) (4,2) (5,2) (6,2)
(1,3) (2,3) (3,3) (4,3) (5,3) (6,3)
(1,4) (2,4) (3,4) (4,4) (5,4) (6,4)
(1,5) (2,5) (3,5) (4,5) (5,5) (6,5)
(1,6) (2,6) (3,6) (4,6) (5,6) (6,6)
(
K
P(F) = ______
P(G) = ______
P(FhÍ1whŒvÜ5?hµ ¿hŒvÜ5?h/¾hŒvÜ5?hÍ1whŒvÜ5?>*[pic] hŒvÜ5?hŒvÜhµ ¿h [pic]Îh/¾h/¾h/¾5?CJ aJ h mÚ5?CJ aJ h/¾5?CJ aJ hiVQh.dZhL=§5?CJ0\?aJ0h³/,5?CJ \?aJ h³/,5?CJ0\?aJ0h³/Equation.DSMT4 [pic]G) =_______
P(F|G) = ________
(
A
Y R
R B
( ( ( (
K K K K
Q Q Q Q
J J J J
10 10 10 10
9 9 9 9
8 8 8 8
7 7 7 7
6 6 6 6
5 5 5 5
4 4 4 4
3 3 3 3
2 2 2 2
A A A A
[pic]
( ( ( (
K K K K
Q Q Q Q
J J J J
10 10 10 10
9 9 9 9
8 8 8 8
7 7 7 7
6 6 6 6
5 5 5 5
4 4 4 4
3 3 3 3
2 2 2 2
A A A A
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- minnesota state employee portal
- minnesota state board of education
- minnesota state department of education
- minnesota state employee self serve
- minnesota state self service portal
- minnesota state teaching license
- minnesota state teacher licensing
- minnesota state music standards
- minnesota state department of licensing
- minnesota state licensing rule 3
- minnesota state licensing board
- minnesota state licensing website