Sets-1-Quiz-1

SETS-1 Quiz-1

Dear Readers,

In mathematics a set is a collection of distinct elements. The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets. Two sets are equal if and only if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century..

Q1. For any two sets A and B, A-(A-B) equals

•  A
•  A-B
•  A∩B
•  A^C∩B^C

Solution
(c) Now, A-(A-B)=A-(A-B^C) =A〖∩(A∩B^C )〗^C =A∩(A^C∪B) =(A∩A^C )∪(A∩B) =A∩B

Q2.If A and B are two given sets, then A∩(A∩B)^c is equal to

•  A
•  B
•  Φ
•  A∩B^c

Solution
(d) We have, A∩(A∩B)^c=A∩(A^c∪B^c) ⇒A∩(A∩B)^c=(A∩A^c)∪(A∩B^c) ⇒A∩(A∩B)^c=ϕ∪(A∩B^c )=A∩B^c

Q3.  Let Z denote the set of integers, then {x∈Z:|x-3|<4}n{x∈Z:|x-4|<5}=

•   {-1,0,1,2,3,4}
•  {-1,0,1,2,3,4,5}
•  {0,1,2,3,4,5,6}
•  {-1,0,1,2,3,5,6,7,8,9}

Solution
(c) We have, {x∈Z:|x-3|<4}={x∈Z:-1

Q4.  The relation ‘is not equal to’ is defined on R, is

•  Reflexive only
•  Symmetric only
•  Transitive only
•  Equivalence

Solution
(b) For any a,b∈R a≠b⇒b≠a⇒R is symmetric Clearly, 2≠-3 and -3≠2, but 2=2. So, R is not transitive. Clearly, R is not reflexive

Q5. Let L denote the set of all straight lines in a plane. Let a relation R be defined by α R β⇔α⊥β,α,β∈L. Then R is

•  Reflexive
•  Symmetric
•  Transitive
•  None of these

Solution
 Symmetric

Q6. An investigator interviewed 100 students to determine the performance of three drinks milk, coffee and tea. The investigator reported that 10 students take all three drinks milk, coffee and tea; 20 students take milk and coffee, 30 students take coffee and tea, 25 students take mile and tea, 12 students take milk only, 5 students take coffee only and 8 students take tea only. Then, the number of students who did not take any of the three drinks, is

•  10
•  20
• 25
•  30

Solution
B

Q7. If n(A×B)=45, then n(A) cannot be

•  15
•  17
•  5
•  9

Solution
(b) We have, n(A×B)=45 ⇒n(A)×n(B)=45 ⇒n(A) and n(B) are factors of 45 such that their product is 45 Hence, n(A) cannot be 17

Q8. Let A and B have 3 and 6 elements respectively. What can be the minimum number of elements in A∪B?

•  3
•  T6
•  9
•  18

Solution
(b) A∪B will contain minimum number of elements if A⊂B and in that case, we have n(A∪B)=n(B)=6

Q9. If A,B and C are three sets such that A⊃B⊃C, then (A∪B∪C)-(A∩B∩C)=

•  A-B
•  B-C
•  A-C
•  None of these

Solution
(c) We have, A⊃B⊃C ∴A∪B∪C=A and A∩B∩C=C ⇒(A∪B∪C)-(A∩B∩C)=A-C

Q10. Let A={ONGC,BHEL,SAIL,GAIL,IOCL} and R be a relation defined as “two elements of A are related if they share exactly one letter”. The relation R is

•  Anti-symmetric
•  Only transitive
•  Only symmetric
• Equivalence

Solution
(c) Clearly, R={(BHEL,SAIL),(SAIL,BHEL),(BHEL,GAIL), (GAIL,BHEL),(BHEL,IOCL),(IOCL,BHEL)} We observe that R is symmetric only