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# Class Note for MATH 1313 at UH

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This 13 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 36 views.

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Date Created: 02/06/15

Section 61 Sets and Set Operations Section 61 Sets and Set Operations A collection of objects is called a An object of a set is called an Notation e quotelement ofquot 65 quotnot an element ofquot Example 1 Let B a b c y z In setbuilder notation the set B can be written as follows Equality of Sets Let A and B be two sets We say that A is equal to B written as A B This is true if and only ifA and B have exactly the same elements If two sets are not equal we write A B Subsets Let A and B be two sets We say that A is a subset ofB or that is contained in B and written A EB From the de nition it follows that for any set A A EA that is every set is a subset ofitself Proper Subsets IfA EB butA B then A is a proper subset of B IfA is a proper subset of B then we write A CB In other words A is a proper subset of B if the following two conditions hold 1 A EB 2 There exist at least one element in B that is not in A Example 2 LetA 123 B 12345and C 321 In the following answer true or false in the following AC TorF AEC TorF ACB TorF CCA TorF A set that contains no elements is called the Note We write 0 to denote the empty set The symbol 0 is a subset of every set Section 61 Sets and Set Operations Example 3 LetA a b c List all subsets of the set A The Universal set is the set ofinterest in a particular discussion A Venn diagram is a Visual representation ofsets They look like Aquot Set Operations Set Union Let A and B be two sets The set of all elements that that belong to either A or B or both is called the ofA and B denoted A U B In set builder notation A U B Set Union in a Venn diagram looks like Section 61 Sets and Set Operations Set Intersection Let A and B be two sets The set of all elements in common with both sets A and B is called the ofA and B denoted AnB In setbuilder notation AnB Set Intersection in a Venn diagram looks like IfA n B 0 then we say the intersection is the null intersection and that A and B are disjoint Set Complement Let U be a universal set and A E U The set of all elements in U that are not in A is called the ofA In setbuilder notation AC Set Complementation in a Venn diagram looks like Section 61 Sets and Set Operations Set Complementation If U is a universal set and A E U then aUCZ bZCU cACCA dAUACU eAn ACZ Set Operations Let A B and C be subsets of U aAUBBUA dAnBncAnBnc bAnBAnB eAuBnCAuBnAuC cAuBucAu BU c fAnBUCAn BUAn C DeMorgan39s Laws aAu BCACnBC bAnBCACUBC Example 3 Let U12345678910 A13579 B246810 C12458 Find the given sets a A U B bBnC Section 61 Sets and Set Operations c BC dBnCC eAUBUCC fAUBCn C g AC 130 u C h cc u B n AC Example 4 Let U denote the set of all employees at a certain Company Let VXe U X likes to read Vogue magazine EXE U X likes to read The Enquirer and RXE U X likes to read Reader s Digest Part A Describe the given set in words given statement in set notation 139 V U R the set of all employees at this company that Section 61 Sets and Set Operations c BC dBnCC eAUBUCC fAUBCn C g AC 130 u C h cc u B n AC Example 4 Let U denote the set of all employees at a certain Company Let VXe U X likes to read Vogue magazine EXE U X likes to read The Enquirer and RXE U X likes to read Reader s Digest Part A Describe the given set in words given statement in set notation 139 V U R the set of all employees at this company that Section 61 Sets and Set Operations 139139 VC n E U R the set of all employees at this company that Part B Describe the given statement in set notation 139 The set of all employees at this company that like Vogue and do not like The Enquirer ii The set of all employees at this company that do not like Vogue Reader s Digest or The Enquirer Another good example is example 14 in your book Read through that example Example 5 Shade the portion of the Venn diagram that represents the given set Assume the given sets are not disjoint a A n BC than Section 61 Sets and Set Operations cBnCCnAC dACnBUC 6 cc u BC UACC Section 62 The Number of Elements in a Finite Set Section 62 The Number of Elements in a Finite Set Let A be a set then nA is the Example 1 LetA 1 2 3 19 20 and B q s t V Find a nA b nB c n03 Given two sets A and B 1 lfA and B are disjoint then nA U B nA nB 21fA and B are not disjoint then nA U B nA nB nA n B Example 2 Let G12345 H246 I789 find anG U H bnH u 1 cnG n H Example 3 LetA and B be subsets ofa universal set U Given that nB 9 nA n B 5 and nA U B 20 find nA Section 62 The Number of Elements in a Finite Set Example 4 LetA and B be subsets ofa universal set U Given that nU 100 nA 61 nB 56 and nA U BC 30 Find 71AC n BC Example 5LetnU 76nA 45nB 40 nC 41nAn B 24 nB n C 22 nA n C 30 and nA n B n C 16 Find the number in each 0fthe following sets Aquot a nA U B n C b B n Ccl Section 62 The Number of Elements in a Finite Set Example 6 In a survey of 374 coffee drinkers it was found that 227 take sugar 245 take cream and 163 take both sugar and cream with their coffee How many take sugar or cream but not both Example 7 In a consumer survey 67 students were asked about which subject they enjoy most The survey revealed that 33 enjoy Math 45 enjoy Science 40 enjoy English 17 enjoy Science and English only 16 enjoy both Math and Science 14 enjoy all three subjects and 25 enjoy exactly two of the three subjects a How many students surveyed enjoy Math and English b How many students surveyed enjoy Science or English but not Math c How many students surveyed enjoy at most one of these three subjects mentioned Section 63 Generalized Multiplication Principle Section 63 Generalized Multiplication Principle Suppose a task T1 can be performed in N1 ways a task T2 can be performed in N2 ways and finally a task Tn can be performed in Nl ways Then the number of ways of performing the tasks T1 T2 Tn in succession is given by the product N1 N239 39Nquot Example 1 A coin is tossed 3 times and the sequence of heads and tails is recorded a Determine the number of outcomes of this activity b List the outcomes of this experiment by rst drawing a tree diagram Example 2 The Burger Bar offers the following items on its menu Burger Sides Beverages Desserts Single Meat Fries Tea Cheesecake Double Meat Onion Rings Coffee Brownie Fruit Bowl Soda Cookie Cheddar Peppers Ice Cream Cone lfa customer chooses 1 item from each category how many meals can be made List 1 meal possible Section 63 Generalized Multiplication Principle Example 3 An identi cation number for employees at a certain company contains six digits How many ID numbers are possible if repetition is allowed Example 4 A license plate consists of 2 letters followed by 4 digits How many license plates are possible if the 1st letter can t be 0 the 1st digit can t be 0 and no repetitions are allowed Example 5 In the original plan for area codes in 1945 the first digit could be any number from 2 through 9 the second digit was either 0 or 1 and the third digit could be any number except 0 With this plan how many different area codes were possible Example 6 Six performers are to present their comedy acts on a weekend evening at a comedy club One of the performers insists on being the last standup comic of the evening If this performer s request is granted how many different ways are there to schedule the appearances Example 7 The call letters for radio station begin with K or W followed by 3 additional letters How many sets of call letters haVing 4 letters are possible Repetition is allowed

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