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1 Name: Sets set is a well-defined collection of distinct objects. Well-defined means that we can determine whether an object is an element of a set or not. Distinct means that we can tell the objects apart. Basic Set Definitions Well-defined Set The set of natural numbers between 3 and 7, inclusive: {1, 2, 3, 4, 5, 6, 7}. { a } The set of rational numbers: b a, b are integers, b 0. Not Well-defined Set The set of large numbers. Set with Non-distinct Objects {, B, C, C} niversal set - In a given problem, the single larger set from which all the elements of the sets under discussion are drawn. Empty set or Null set - or {} set containing no elements. Equivalent Sets Sets and B are equivalent if and only if n() = n(b). One-to-One Correspondence If two sets have the same cardinality, they can be put in a one-to-one correspondence by matching each element in one set with a different element in the other set. Venn diagram tool used in visualizing sets invented by John Venn ( ). B

2 Set definitions Definition Example is an element of 2 {0, 1, 2, 3, 4, 5, 6} / is not an element of 2 / {1, 3, 5} is a subset of {1, 3, 5} {0, 1, 2, 3, 4, 5, 6, 7, 8} {1, 3, 5} {1, 3, 5} is not a subset of {0, 1, 2, 3, 4, 5, 6} {1, 3, 5} is a proper subset of {3, 5} {1, 3, 5} is not a proper subset of {1, 3, 5} {1, 3, 5} = set equality {1, 3, 5} = {3, 1, 5} The order of elements does not matter Number of Subsets of Set If n() = k, then has 2 k subsets and 2 k 1 proper subsets. 1. Determine whether is a subset of B. Draw a Venn diagram for each case. (a) = {2, 3, 4} and B = {1, 2, 3, 4, 5} (b) = {0, 2, 3, 4} and B = {1, 2, 3, 4, 5} The Complement and Set Definitions involving Two Sets For the examples in the table below, suppose set = {0, 2, 4, 6, 8}, set B = {1, 3, 5, 7, 9}, set C = {1, 2, 3, 4} and the universal set, = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Set definitions Definition Example The intersection of two sets B = and B = the set of C = {2, 4} all elements in both and B The union of two sets and B = C = {0, 1, 2, 3, 4, 6, 8} the set of all elements B = in either or B or both The complement of a set, C = {0, 5, 6, 7, 8, 9} = all elements in the = {1, 3, 5, 7, 9} = B niversal set,, not in set Two sets, and B are disjoint, and B are disjoint, if B =. since B = B and C are not disjoint, since B C = {1, 3} Page 2

3 The Number of Elements in a Set - Cardinality n() denotes the number of elements in, if is a finite set. 2. Calculate n() for each set below. (a) = {1, 2, π, e, 11, 4 123}, n() = (b) = the letters of the alphabet, n() = (c) =, n() = Important Counting Formulas The Inclusion-Exclusion Principle 1. n( B) = n() + n(b) n( B), or n( or B) = n() + n(b) n( and B) The Complement Principle 2. n() = n() n( ) 3. Verify the counting formulas above using the Venn diagram below. Each element is represented by a diamond. B (a) n( B) = n() = n(b) = n( B) = (b) n() = n() = n( ) = Page 3

4 Set Difference Law: B = B Double Complement Law: ( ) = Cartesian Product: The cartesian product of a set and set B, B, cross B is the set of all ordered pairs, (a, b) with a and b B. 4. For the sets below, find the set find the set B: (a) = {red, blue, white}, and B = {Ford, Toyota} (b) = {x, y, z}, and B = {0, 1, 2} Page 4

5 5. For the sets = {a, b, c, d, e, g}, B = {c, d, e, f, h, i}, C = {e, g, h, j, l} and the universal set, = {a, b, c, d, e, f, g, h, i, j, k, l, m}, se a Venn diagram to illustrate the three sets above. Then find: (a) C (b) ( B) (c) B (d) B (e) C 6. By labeling regions in a Venn diagram, show that: ( B) = B. B Set Region Set Region Labels Labels B B B ( B) B De Morgan s Laws ( B) = B ( B) = B Page 5

6 7. By labeling regions in a Venn diagram, shade the following set: ( B) C. B C Set B ( B) C ( B) C Region Labels Page 6

7 8. Is (B C) = ( B) ( C)? By labeling regions in a Venn diagram, check if both sets are represented by the same regions or not. B C /09/10 50 Set Region Set Region Labels Labels B B C C (B C) ( B) ( C) 9. se set statements to write descriptions of the shaded areas. se union, intersection and complement as necessary. More than one answer may be possible. Page 7

8 10. Landscape Purchases gway Lawn and Garden collected the following information regarding purchases from 130 of its customers. 74 purchased shrubs. 70 purchased trees. 41 purchased both shrubs and trees. Let S be the set of customers who purchased shrubs and T be the set of customers who purchased trees. se the Venn diagram to answer the questions below. Of those surveyed, (a) how many purchased only shrubs? (b) how many purchased only trees? (c) how many did not purchase either of these items? Page 8

9 11. Electronic Devices In a survey of college students, it was found that 356 owned an ipod. 293 owned a laptop. 285 owned a gaming system. 193 owned an ipod and a laptop. 200 owned an ipod and a gaming system. 139 owned a laptop and a gaming system. 68 owned an ipod, a laptop, and a gaming system. 26 owned none of these devices. Let I be the set of students who owned an ipod, L be the set of students who owned a laptop, and let G be the set of students who owned a gaming system. se the Venn diagram to answer the questions below. Page 9

10 Of those surveyed, (a) How many college students were surveyed? Of the college students surveyed, how many owned (b) an ipod and a gaming system, but not a laptop? (c) a laptop, but neither an ipod nor a gaming system? (d) exactly two of these devices? (e) at least one of these devices? Page 10

11 12. Let the universal set, be the set of all Cabrillo College students, let E be the set of all Cabrillo College students enrolled in an English class, let G be the set of all Cabrillo College students enrolled in a geology class, and let M be the set of all Cabrillo College students enrolled in a math class. (a) se the given set symbols to construct a Venn diagram identifying the various sets of students. (b) se the given set symbols along with, or, to identify the set of students enrolled in English, but not in geology or math. 13. Show that the set is infinite by placing it in a one-to-one correspondence with a proper subset of itself. Be sure to show the pairings of the general terms of the sets. (a) {30, 31, 32, 33, 34,...} (b) { 6 13, 7 13, 8 13, 9 13,...} 14. Show that the set has cardinal number ℵ 0 by establishing a one-to-one correspondence between the set of counting numbers and the given set. Be sure to show the pairings of the general terms of the sets. (a) {0, 2, 4, 6, 8,...} (b) { 1 2, 2 3, 3 4, 4 5,...} Page 11

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