Sets-1-Quiz-16

SETS-1 Quiz-16

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 A={x,y}, then the power set of A is

•  {x^y,y^x}
•  {ϕ,x,y}
•  {ϕ,{x},{2y}}
•  {ϕ,{x},{y},{x,y}}

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

Q2.The finite sets A and B have m and n elements respectively. if the total number of subsets of A is 112 more than the total number of subsets of B, then the volume of m is

•  7
•  9 >
•   10
•  12

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.  Which of the following is an equivalence relation?

•   Is father of
•  Is less than
•  Is congruent to
•  Is an uncle of

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 A={(x,y):y=e^x,x∈R}, B={(x,y):y=e^(-x),x∈R}. Then,

•  A∩B=ϕ
•  A∩B≠ϕ
•  A∪B=R^2
•  None of these

Solution
B

Q5.If A={(x,y):x^2+y^2=4; x,y∈R} and B={(x,y):x^2+y^2=9;x,y∈R}, then

•  A-B=ϕ
•  B-A=B
•  A∩B≠ϕ
•   A∩B=A

Solution
 (d) Clearly, R is an equivalence relation

Q6. The relation R defined in N as a R b⟺b is divisible by a is

•  Reflexive but not symmetric
•  Symmetric but not transitive
• Symmetric and transitive
•  None of these

Solution
(d) Clearly, R is an equivalence relation

Q7. If A,B and C are three sets such that A∩B=A∩Cand A∪B=A∪C, then

•  A=C
•  B=C
•  A∩B=ϕ
•  A=B

Solution
B

Q8. The relation ‘is subset of’ on the power set P(A) of a set A is

•  Symmetric
•  Anti-symmetric
•  Equivalence relation
•  None of these

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

Q9. The number of elements in the set {(a,b):2a^2+3b^2=35,a,b∈Z}, where Z is the set of all integers, is

•  2
•  4
•  8
•  12

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. Let R be a relation on the set N of natural numbers defined by nRm⇔n is a factor of m(i.e. n ┤| m). Then, R is

•  Reflexive and symmetric
•  Transitive and symmetric
•  Equivalence
• Reflexive, transitive but not symmetric

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