70 CHAPTER 2 Sets for purchase by Alexander and Benjamin, respectively. The trading abbreviations of the cryptocurrencies are given in the universal set below. U A B {Bitcoin (BTC), Ethereum (ETH), Tether (USDT), Binance (BNB), U.S. Dollar Coin (USDC), XRP (XRP), Cardano (ADA), Dogecoin (DOGE), Solano (SOL), Polygon (MATIC)} {BTC, USDT, XRP, DOGE, SOL} {ETH, USDT, BNB, SOL, MATIC} = = = Use these sets and the abbreviations given in the universal set to determine the following: a) ′ A B ( ) < {USDC, ADA} b) ′ ′ A B < {BTC, ETH, BNB, USDC, XRP, ADA, DOGE, MATIC} 88. Yoga Poses In this exercise, we use recommendations from the yoga school Raj Yoga Rishikesh to form the universal set, U, that contains the top 10 yoga poses for beginners. Chanda and Diya are yoga instructors. Set C and set D are sets that contain the 5 yoga poses used by Chanda and Diya in their yoga classes respectively. The abbreviations of the poses are given in the universal set below. U C D {Mountain Pose (MP), Tree Pose (TP), Triangle (T), Warrior (W), Downward-Facing Dog (DFD), Upward-Facing Dog (UFD), Seated Forward Fold (SFF),Bridge Pose (BP), Child Pose (CP), Savasana (S)} {T, W, UFD, CP, S} {MP, T, DFD, UFD, S} = = = Use these sets and the abbreviations given in the universal set to determine the following: a) ′ C D ( ) > {MP, TP, W, DFD, SFF, BP, CP} b) ′ ′ C D> {TP, SFF, BP} 89. Gardening The data obtained from a survey of customers at Tractor Supply Company showed that 27 were planting vegetables, 16 were planting flowers, and 11 were planting both vegetables and flowers. How many customers were planting either flowers or vegetables? 32 customers 90. Gaming The data obtained from a survey of senior students at the Academy for the Arts showed that 56 played the game Telestrations, 68 played the game Sequence, and 29 played both games. How many students played either Telestrations or Sequence? 95 students 91. Exercising The data obtained from a survey of members of Platinum Gym showed that 105 members used the gym for weight training or for cardio training, 75 members used the gym only for weight training, and 33 members used the gym for both weight training and cardio training. How many members used the gym only for cardio training? 63 members 92. Sweeteners The data obtained from a survey of customers at Caribou Coffee showed that 321 customers used either stevia or monk fruit extract to sweeten their coffee. Of these customers, 27 used both stevia and monk fruit extract and 89 used only monk fruit extract. How many customers used only stevia to sweeten their coffee? 259 customers Challenge Problems/Group Activities 93. When will a set and its complement be disjoint? Explain and give an example. A set and its complement will always be disjoint. For example, if = = U A {1, 2, 3} and {1, 2}, then ′ = A {3}, and ′ = A A {}. > 94. When will = n A B ( ) 0? > Explain and give an example. n A B ( ) 0 > = when A and B are disjoint sets. A B n A B {1, 3}, {2, 4}, ( ) 0 > = = = 95. Consider the formula = + − nAB nAnBnAB ( ) ( ) ( ) ( ) < > a) Show that this relation holds for = A a b c d {, , , }and = B b d e f g h {, , , , , }. = + − 8 4 6 2 b) Make up your own sets A and B, each consisting of at least six elements. Using these sets, show that the relation holds. Answers will vary. c) Use a Venn diagram and explain why the relation holds for any two sets A and B. Answers will vary. 96. The Venn diagram in Fig. 2.14 shows a technique of labeling the regions to indicate membership of elements in a particular region. Define each of the four regions with a set statement. (Hint: ′ A B > defines region I.) I II III IV U A x [ A x [ A9 and x [ B9 x [ B9 and x [ A9 x [ B and x [ A x [ B and B Figure 2.14 In Exercises 97–104, let = … U {0, 1, 2, 3, 4, 5, }, = A {1, 2, … 3, 4, }, = … B {4, 8, 12, 16, }, and = … C {2, 4, 6, 8, }. Determine the following. 97. A B < … {1, 2, 3, 4, }, or A 98. A B > … {4, 8, 12, 16, }, or B 99. B C< … {2, 4, 6, 8, }, or C 100. B C> … {4, 8, 12, 16, }, or B I A B > ′ II A B > III A B >′ IV A B >′ ′ or A B ( ) < ′ Rawpixel/123RF
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