Question

1. U = {1, 2, 3, 4, 5, … 10} A = {6,7,8,9,10} B = {1, 3, 5, 7, 9}, C = { 2, 4, 6, 8, 10} b) Find B’ c) Find A ∪ (B ∩ C)

2. Use Venn diagrams to show why A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

3. In a large class, there are 45 students who are either on a sports team or involved in an academic club (such as Forensics team). There are 22 students on a sports team and 27 students in an academic club. How many are both on a sports team and an academic club.

4. A city offers three types of mass transit: buses, subways, and trains. A survey of 200 city residents found the following 73 did not use any mass transit at all 82 used the subway 44 used only the subway 28 used both the bus and the subway 59 used the train. 10 people used all three types of transit 15 people used both the train and the bus. Draw a Venn diagram with circles for B (bus), S (subway) and T(train). Fill in the numbers. How many people in the survey used only the train? How many used only one type of mass transit (only bus or only train or only subway)?

Answer 1

1. U = {1, 2, 3, ...10} A={6,7,8,9, 10} , B={1,3,5,7,9} c=22,4,6,8,103 :.nco=10, n(A)=5; n(B) = 5, DOOS b> B'= {x12EU but x&B} = {2,4,6,9,10} .ncb')=5 - PCB)= n(BP) - PCB) = {] e) (Bnc)={X/XEB and Rech ={ } .: AUCBnc) = {n/HEA or reconc)} ={6,7,8,9,10% in [Au(Bach) 5 .P[auceno)] = n(AUCBOO] s neu) To nou 's

2) Venn diagram of An(Bue) X AN(BUC) Ano ANB B AMBUCAN vern diagram of CANBUCAnc) cares TU Both renn diagram represents the same region. :: AN (BUC) = CANBUCane) 3) let À Be the event of students who belong to sports team. let 'B' be the event of students who involved in an academic club. i from given information hp(AUB) = 45 np (A)=22 & n(B)=27 ... The number of students who are both

on a sports tean and an academic club is given by ncanB)= n(A) + n(B)- n(AUB) = 22+27-45 =49-45 = 4 CANB) = 47. 4. Let B (bus), s(subway) and T(train) - represents three types of transit. giren: ncur=200 73 did not use any mass transit. inCAUBUC)'= 73 82 used subway il nis)=82 28 used both bus and the subway no Brs)=28 59 Used train nCT2=59 to people used all three type осноос) = 10

15 used both train and bus nCTOB)=15 and 44 used only subway venn diagram is nev=200 X4057 1 44 82 (TNB) L >(Bns) 'B 731 The number of people in suerey used only the train is sy obtained by: giren 82 used subway, out of which 44 used only subway. :- 44 82-44=38 usce other transit, also, n(Brs)-nCanon() =18 => nctns)=38-18=20 : The people used only train is 59-(18+15) = 59-33 = 26 ? 26 only used train The number of people used only one type of mass transit is giren by: zoo 127-33=94
since the number of person who use any one of the transits are 200-73=127

And 33 people use more than one transit.

Thus of the number of people use only one type of mass transit are 94