MAT 141 ‐ Chapter 7 Finite Mathematics MAT 141: Chapter 7 Notes Sets Set Theory and Probability David J. Gisch Definition Definition • A set is a well-defined collection of objects. • Repetitions of elements do not matter. Whether it is listed once or twice it is still a member of the set and that ▫ We denote sets with capital letters is all that matters. ▫ We write sets with brackets as follows 3, 4, 5 • Order also does not matter in sets, unless it is used to ▫ This is referred to as roster form of a set. establish a pattern. 3, 4, 5 � 4, 3, 5 � 3, 3, 3, 4, 5 � 5, 5, 3, 4, 4, 4 • Any item belonging to a set is called an elem ent or m em ber of that set. • A set can also contain no elements. We call this the ▫ We denote elements of a set as follows em pty set . 3 ∈ 3, 4, 5 ▫ We denote the empty set as ∅ 7 ∉ 3, 4, 5 ▫ For example, the set of months starting with the letter z. ▫ Note that ∅, 0, 0 , ��� ∅ all have different meanings. Why well-defined? Give me the set of people in this room who are nice. 1
MAT 141 ‐ Chapter 7 Definition S ets • The set of all things being discussed is referred to as the Example 7.1.1: Explain the difference for each of the universal set . We denote the universal set as set � . following. • For example, if we were discussing arithmetic in third ∅ grade we might use the universal set of whole numbers. In college algebra the universal set would be all real numbers. 0 0 ∅ S ets Definition • Sets can also be written in set-builder notation . Example 7.1.2: Write out each of the following sets in roster �|� ∈ � ��� 2 � � � 5 form. The set (a) The set of all numbers between 2 and 7. They are integers and between 2 and 5. Of things Such that In roster form (b) The set days of the week that begin with the letter S. 3, 4 Obviously above it doesn’t help but what about �|� �� � ���� ���� ������ ���� ��� ������ � (c) The set of planets in our solar system that begin with the letter C. 2
MAT 141 ‐ Chapter 7 S et-Builder Notation S et-Builder Notation Example 7.1.3: Write each of the following sets in set- Example 7.1.4: Write each of the following sets roster form. builder notation. �|� �� �� ������� (a) (a) 2, 4, 6, 8, 10, … (b) �3, �2, 0, 1, 2, 3, 4 (b) �|� �� � ����� ���� ������� ���� (c) 15, 16, 17, 18, … . Definition S ubsets • Sometimes every element of set is also the element of Example 7.1.5: List all of the subsets of the set 3, 4 another set. ▫ � � �|� �� � ����� ������� �� ���� ���� ▫ � � �|� �� � ������ �� ���� ���� ▫ Here, every element of set A is also an element of set B. • Think of a proper subset as being strictly smaller. When in doubt do not write � ⊂ � , write � ⊆ � . 3
MAT 141 ‐ Chapter 7 S ubsets Number of S ubsets Example 7.1.5: List all of the subsets of the set �, �, � Example 7.1.5: List all of the subsets of the set �, �, � Number of S ubsets S et R elations Example 7.1.6: How many subsets does the following set • Sets can be have? ▫ Proper subsets (one is contained in the other). ����, �, �, 3, �2 � � �|� �� � ����� � � �|� �� � ������ ▫ Have some overlap (called the intersection). � � �|� �� � ���� ����� � � �|� �� � ��� ������ ▫ Have no overlap (called disjoint sets). � � �|� �� � ������� � � �|� �� � ������ 4
MAT 141 ‐ Chapter 7 S et R elations S et R elations S et R elations S et R elations 5
MAT 141 ‐ Chapter 7 S ets, Union, Intersection, Complement S ets, Union, Intersection, Complement Example 7.1.7: Use the given sets to state each of the Example 7.1.8: Use the given sets to state each of the following in roster form. following in roster form. � � �, �, �, �, �, �, �, � (a) � ∪ � (a) � ∪ � � � �, �, � � � �, �, � (b) � ∩ � � � ��, �� (b) � ∩ � (c) � ∪ ∅ (c) B ′ (d) �′ (d) �′ ∩ �′ (e) �′ ∩ � (e) �� ∪ ��′ (f) �� ∩ ��′ Venn Diagram (2 S ets) • 2 sets split the diagram up into 3-4 regions. Applications of Venn Diagrams 6
MAT 141 ‐ Chapter 7 Venn Diagrams Venn Diagrams Example 7.2.1: Write each shaded region using set notation. Example 7.2.2: Write each shaded region using set notation. Venn Diagram (3 S ets) Venn Diagrams • 3 sets split the diagram up into at most 8 regions. Example 7.2.3: Write each shaded region using set notation. 7
MAT 141 ‐ Chapter 7 Venn Diagrams Venn Diagrams Example 7.2.4: Assume that A: Set of athletes, B: Set of Example 7.2.5: Write each indicated region using set honors students, and C: Set of Band students. Describe each notation. of the following regions in words. II II IV and V IV and V I and II and III I and II and III S ets, Union, Intersection, Complement Making Venn Diagrams Example 7.2.6: Use the given sets to state each of the • Peel your way out!!!! following in roster form. ▫ Start with the inner-most region first. ▫ Go to the intersections and subtract off what you already have. (a) � ∪ � ∪ � � � �, �, �, �, �, �, �, �, �, � ▫ Go to the remainder of the sets and subtract off what you already � � �, �, �, �, � have. � � �, �, �, �, � � � ��, �� ▫ Always check if any amount is unused. (b) � ∩ � ∩ � • For example: Let’s say there are 20 total elements and A has 12 elements, B has 10 elements and A intersect B has (c) � ∪ � ∩ � 8 elements. (d) � ∪ � ∩ �� ∪ �� � ∩ �′ ∩ �� ∪ � � � (e) 8
MAT 141 ‐ Chapter 7 The Number of Elements Venn Diagrams Example 7.2.7: Eight hundred students were surveyed and the results of the campus blood drive survey indicated that 490 students were willing to donate blood,340 students were willing to help serve a free breakfast to blood donors, and 120 students were willing to donate blood and serve breakfast. (a) How many students were willing to donate blood or serve breakfast? (b) How many were willing to do neither? Venn Diagrams Venn Diagrams Example 7.2.8: A survey of 120 college students was taken at Example 7.2.9: Use the Venn Diagram below to answer the following. registration. Of those surveyed, 75 students registered for a math course, 65 for an English course, and 40 for both math and English. Of (a) How many students read none of the publications? those surveyed, (b) How many read Business Week and Fortune but not the Journal ? (a) How many registered only for a math course? (b) How many registered only for an English course? (c) How many read Business Week or the Journal ? (c) How many registered for a math course or an English course? (d) How many read all three? (d) How many did not register for either a math course or an English course? (e) How man do not read the Journal? 9
MAT 141 ‐ Chapter 7 Venn Diagrams Example 7.2.10: A survey of 180 college men was taken to determine participation in various campus activities. Forty-three students were in fraternities, 52 participated in campus sports, and 35 participated in various campus tutorial programs. Thirteen students participated in fraternities and sports, 14 in sports and tutorial programs, and 12 in fraternities and tutorial programs. Five students participated in all three activities. Create the Venn Diagram for this scenario. Introduction to Probability Definitions Theoretical Method for Equally Likely Outcomes • An experim ent is an activity or occurrence with an Count the total number of possible Step 1: observable result. outcomes. Among all the possible outcomes, count Step 2: • Outcom es are the most basic possible results of the number of ways the event of interest, observations or experiments. E , can occur. • The set of all possible outcomes of an experiment is Determine the probability, ���� . Step 3: called the sam ple space . ������ �� ���� � ��� ����� • An event consists of one or more outcomes that share a � � � property of interest. Or think of an event as a subset of ����� ������ �� �������� �������� the sample space. � � � ���� ���� 10
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