Sets-1-Quiz-17

SETS-1 Quiz-17

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. If n(A∩B)=5,n(A∩C)=7 and n(A∩B∩C)=3, then the minimum possible value of n(B∩C) is

•  0
•  1
•  3
•  2

Solution
(c) Minimum possible value of n(B∩C) is n(A∩B∩C)=3

Q2.The relation “is a factor of” on the set N of all natural numbers is not

•  Reflexive
•  Symmetric >
•   Antisymetric
•  Transitive

Solution
(b) Clearly, 2 is a factor of 6 but 6 is not a factor of 2. So, the relation ‘is factor of’ is not symmetric. However, it is reflexive and transitive

Q3.  If A_1,A_2,…,A_100 are sets such that n(A_i )=i+2,A_1⊂A_2⊂A_3…⊂A_100 and ⋂_(i=3)^100▒〖A_i=A,〗 then n(A)=

•   3
•  4
•  5
•  6

Solution
(c) It is given that A_1⊂A_2⊂A_3⊂⋯⊂A_100 ∴⋃_(i=3)^100▒〖A_i=A⇒A_3=A⇒n(A)=n(A_3 )=3+2=5〗

Q4. Let R be a relation from a set A to a set B, then

•  R=A∩B
•  R⊆A×B
•  R⊆A×B
•  R⊆B×A

Solution
B

Q5.On the set of human beings a relation R is defined as follows: "aRb iff a and b have the same brother”. Then R is

•  Only reflexive
•  Only symmetric
•  Only transitive
•  Equivalence

Solution
 (d) Clearly, R is an equivalence relation

Q6. X is the set of all residents in a colony and R is a relation defined on X as follows: “Two persons are related iff they speak the same language” The relation R is

•  Only symmetric
•  Only reflexive
• Both symmetric and reflexive but not transitive
•  Equivalence

Solution
(d) Clearly, R is an equivalence relation

Q7. Let R and S be two relations on a set A. Then, which one of the following is not true?

•  R and S are transitive, then R∪S is also transitive
•  R and S are transitive, then R∩S is also transitive
•  R and S are reflexive, then R∩S is also reflexive
•  R and S are symmetric, then R∪S is also symmetric

Solution
B

Q8. If n(A)=4,n(B)=3,n(A×B×C)=240, then n(C) is equal to

•  288
•  1
•  12
•  2

Solution
(d) ∵ n(A×B×C)=n(A)×n(B)×n(C) ∴ n(C)=24/(4×3)=2

Q9. The void relation on a set A is

•  Reflexive
•  Symmetric and transitive
•  Reflexive and symmetric
•  Reflexive and transitive

Solution
(b) The void relation R on A is not reflexive as (a,a)∉R for any a∈A. The void relation is symmetric and transitive

Q10. For any two sets A and B, if A∩X=B∩X=ϕ and A∪X=B∪X for some set X, then

•  A-B=A∩B
•  A=B
•  B-A=A∩B
• None of these

Solution
(b) Given, A∩X=B∩X=ϕ ⇒A and X,B and X are disjoint sets. Also, A∪X=B∪X⇒A=B