Unit 1 sets lecture notes

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Unit 1 sets lecture notes

  1. 1. Unit 1 Sets §2.1: Symbols and Terminology ..........................................................................................3 §2.2: Venn Diagrams and Subsets .......................................................................................7 §2.3: Set Operations ...........................................................................................................13 §2.4: Surveys and Cardinal Numbers ................................................................................21
  2. 2. §2.1: SYMBOLS AND TERMINOLOGY Definition: A set is a ______________ of objects. If an object belongs to the set, we call it an ______________ or ______________. Notation: x ∈ A means x is ____ ___________ of A. x ∉ A means x is ____ ____ ___________ of A. Remark A set must be ______________________. This means that we can determine if an element belongs to the collection. Naming Sets There are three ways to name a set: • Listing (Roster Form) The elements of the set are listed, separated by ______________. The entire list is enclosed in braces, • { }. The order of the elements is not important. Word Description (Verbal Description) The description provided must be ______________. • Set-builder Notation Set-builder notation encloses a general ______________ for the elements and a description of any restrictions on the formula input. { formula for element | restriction on formula's input} Example 1: Write a verbal description of the set. A = {2,3,5, 7,11,13,17,19} Solution: Unit 1: Sets 3
  3. 3. Example 2: List the elements in the set. B = { x 2 − 1| x ∈ N and x ≤ 5 } Solution: Example 3: Identify each collection as “well defined” or “not well defined”. A. { x | x is a book catalogued by JALC} B. { x | x is a good movie} C. { x | x is a counting number greater than 4} Solution: A. B. C. 4 Unit 1: Sets
  4. 4. Example 4: List the elements of the set. A is the set of odd counting numbers greater than 10. Solution: Example 5: List the elements of the set. A is the set of all astronauts who have walked on Neptune. Solution: Two Special Sets Definition: The ____________ ______ is the set that contains no elements. Notation: We can write _____ or _______. Definition: The ________________ ________ is the set that contains all of the elements relevant to the context. Notation: We write _______. Unit 1: Sets 5
  5. 5. Equal Sets Definition: Two sets are said to be equal if they have the __________ ________________. Notation: If A and B are equal, then we write A = B. Cardinality of a Set Definition: The ______________ of _________________ in a set is called the cardinal number of a set. Notation: n ( A ) = cardinal number of A Example 6: Find n ( A ) . A = { x | x is a vowel in the English alphabet} Solution: Example 7: Find n ( B ) . B = { x | x is two-digit natural number} Solution: 6 Unit 1: Sets
  6. 6. §2.2: VENN DIAGRAMS AND SUBSETS Venn Diagrams Definition: A Venn diagram is a ___________ _______________ of a set or sets. Venn diagrams will be particularly useful when we discuss set operations in §2.3. Example 1: In the Venn diagram below, each region is given a number. U B A 2 1 3 4 A. List the numbered regions which correspond to the set A. B. List the numbered regions which correspond to the set B. C. List the numbered regions which correspond to the set U. Solution: A. B. C. Unit 1: Sets 7
  7. 7. Example 2: Shade the regions in the Venn diagram which correspond to the elements that are not in A. U B A 2 1 3 4 Complement of a Set Definition: The complement of a set A is the set of all elements that are ________ _____ the set A. Notation: A′ = the complement of A. Remark: The regions we shaded in Example 2 correspond to A′. Example 3: Write the definition of the complement of the set A in set-builder notation. Solution: 8 Unit 1: Sets
  8. 8. Example 4: Given U = {rain, snow, sleet, hail, wind, frost, fog} and A = {snow, wind, frost} , find A′. Solution: Example 5: Find each of the following. A. U ′ B. ∅′ C. ( A′)′ Solution: A. B. C. Unit 1: Sets 9
  9. 9. Subsets Example 6: When you order a hamburger at PawPaw J’s Hamburger Grill, you can pick from the following set of condiments. {ketchup, mustard, pickles} Write out sets that correspond to every possible choice of condiments. Solution: Definition: We say the set A is a subset of the set B if ____________ _________________ that belongs to A ___________ _______________ to B. Notation: A ⊆ B means A is a subset of B. Example 7: List all of the subsets of the set {hot, cold, warm, frigid} . Solution: 10 Unit 1: Sets
  10. 10. Formula: Let A be a set with n elements. Then A has 2n subsets. Note: Where did the two come from? Hint: Remember that a set must be well-defined. Example 8: Given A = {gold, silver, bronze, magnesium, copper} , how many subsets does the set A have? Solution: Two Unexpected Subsets Example 9: Let A be a nonempty set. What are two sets that are guaranteed to be subsets of A? Solution: Definition: If A is a subset of B and A ≠ B, we call A a _____________ _______________ of B. Notation: A ⊂ B means A is a proper subset of B. Unit 1: Sets 11
  11. 11. Formula: Let A be a set with n elements. Then A has __________ proper subsets. Example 10: Let T = {Edwards, Owen, Calvin, Luther, Hodge, Augustine} . How many proper subsets of T are there? Solution: 12 Unit 1: Sets
  12. 12. §2.3: SET OPERATIONS Example 1: In the Venn diagram below, shade the region(s) that correspond to the elements that are in both A and B. U B A 2 1 3 4 Example 2: Neldys will be happy with any combination of the following set of toppings for a pizza N = {pepperoni, sausage, mushrooms, olives, peppers} , and Jim will be happy with any combination of the following set of toppings for a pizza J = {sausage, onions, tomatoes, olives, bacon} . With what set of toppings would they both be happy? Solution: Unit 1: Sets 13
  13. 13. Intersections Definition: The intersection of the set A and the set B is the set of those elements that are in both A and B. Notation: A ∩ B means the intersection of A and B. Remark: You will notice that we shaded the region in Example 1 that corresponds to A ∩ B. Example 3: Write the definition of A ∩ B in set-builder notation. Solution: Example 4: Let U = {Freud, Jung, Piaget, Skinner, Bandura, Rogers, Pavlov, Lewin, Erikson, James} , A = {Freud, Piaget, Skinner, Jung} , and B = {Piaget, Jung, Rogers, Lewin} . Find A′ ∩ B. Solution: 14 Unit 1: Sets
  14. 14. Unions Example 5: Debbie rolls a single, six-sided die. She will win a necklace if she rolls an even number or a number less than 3. Write the list of outcomes for which she would win. Solution: Definition: The union of the set A and the set B is the set of those elements that are in A or B (or both). Notation: A ∪ B means the union of A and B. Example 6: Write the definition of A ∪ B in set-builder notation. Solution: Example 7: In the Venn diagram below, shade the region(s) that correspond to A ∪ B. U B A 2 1 3 4 Unit 1: Sets 15
  15. 15. Example 8: Let U = {a, b, c, d , e, f , g , h, i, j} , A = {a, b, d , g , h} , and B = { g , h, i, j} . Find each of the following. A. A′ B. B′ C. A ∪ B D. A ∩ B E. F. G. A′ ∪ B′ H. A′ ∩ B′ ( A ∪ B )′ A. B. C. D. E. F. G. H. 16 ( A ∩ B )′ Unit 1: Sets
  16. 16. Rule: DeMorgan’s Law for Sets • ( A ∪ B )′ =A′ ∩ B′ • ( A ∩ B )′ =A′ ∪ B′ Differences Definition: The difference of the sets A and B is the set of those elements that are in A but not B. Notation: A − B means the difference of A and B. Example 9: Write the definition of A − B in set-builder notation. Solution: Example 10: In the Venn diagram below, shade the region(s) that correspond to A − B. U B A 2 1 3 4 Unit 1: Sets 17
  17. 17. Example 11: Let U = {a, b, c, d , e, f , g , h, i, j} , A = {a, b, d , g , h} , and B = { g , h, i, j} . Find each of the following. A. A − B B. B − A C. U − A D. What is another name for U − A ? A. B. C. D. Warning: 18 A − B does not mean the same thing as B − A. Unit 1: Sets
  18. 18. Shading Venn Diagrams Use the numbered regions to help you determine which regions to shade. Think of it as “paintby-number.” Example 12: In the Venn diagram below, shade the region(s) corresponding to A′ ∩ B. U B A 2 1 3 4 Solution: Unit 1: Sets 19
  19. 19. Example 13: In the Venn diagram below, shade the region(s) corresponding to ( A ∪ B′ ) ∩ C. U B A C Solution: 20 Unit 1: Sets
  20. 20. §2.4: SURVEYS AND CARDINAL NUMBERS Cardinality when Two Sets are Involved Example 1: In a group of 47 students, 22 students are enrolled in a science class, 28 are enrolled in a humanities class, and 7 are enrolled in both. A. How many students are enrolled in a science class or a humanities class? B. How many are enrolled in neither? Solution: Example 2: Find the value of n ( A ∪ B ) = 19, n ( B ) 14, and n ( A ∩ B ) = if n ( A ) = 4. Solution: Unit 1: Sets 21
  21. 21. Developing a Formula for n ( A ∪ B ) U B A 2 1 3 4 In the above Venn diagram, • Determine the regions accounted for in each of the following. o o n ( B ) counts the elements in regions ____________ o n ( A ) + n ( B ) counts the elements in regions ____________ and ________ o • n ( A ) counts the elements in regions ____________ n ( A ∪ B ) counts the elements in regions ____________ Does the formula n ( A ∪ B ) n ( A ) + n ( B ) make sense? Explain why or why not. ______ = ________________________________________________________________________ ________________________________________________________________________ • If we want to count the elements in region 2, what notation would we use? ___________ Formula: For any two sets A and B, n ( A ∪ B) = _______________________________ 22 Unit 1: Sets
  22. 22. Example 3: Find the value of n ( A ∩ B ) = 30, n ( B ) 19, and n ( A ∪ B ) = if n ( A ) = 43. Solution: Cardinality when Three Sets are Involved Example 4: Use the given Venn diagram and the given information to fill in the number of elements in each region. n ( A ∩ B ) 21, n ( A ∩ B ∩ C ) 6, n ( A ∩ C ) 26, n ( B ∩ C ) 7, = = = = n ( A ∩ C ′ ) 20, n ( B ∩ C ′ ) 25, n (= 40, and n ( A′ ∩ B′ ∩ C ′ ) 2. C) = = = U B A C Unit 1: Sets 23
  23. 23. Example 5: A survey of 80 movie renters was taken. 25 45 32 10 12 7 4 enjoy horror films. enjoy romantic comedies. enjoy documentaries. enjoy horror films and romantic comedies. enjoy romantic comedies and documentaries. enjoy horror films and documentaries. enjoy all three. Use a Venn diagram to answer each question. A. How many enjoy documentaries and romantic comedies, but not horrors? B. How many enjoy none of these of three types of movies? 24 Unit 1: Sets

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