Sets-1-Quiz-3

SETS-1 Quiz-3

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.  In a battle 70% of the combatants lost one eye, 80% an ear, 75% an arm, 85% a leg, x% lost all the four limbs. The minimum value of x is

•  10
•  12
•  15
•  None of these

Solution
(a) 10

Q2.If n(A) denotes the number of elements in the set A and if n(A)=4,n(B)=5 and n(A∩B)=3, then n[(A×B)∩(B×A)] is equal to

•  8
•  9
•  10
•  11

Solution
(b) Given, n(A)=4, n(B)=5 and n(A∩B)=3 ∴n[(A×B)∩(B×A)]=3^2=9

Q3.  If R={(a,b):|a+b|=a+b} is a relation defined on a set {-1,0,1}, then R is⊂

•   Reflexive
•  Symmetric
•  Anti symmetric
•  Transitive

Solution
(b) Let (a,b)∈R. Then, |a+b|=a+b⇒|b+a|=b+a⇒(b,a)∈R ⇒R is symmetric

Q4.  Let n(U)=700,n(A)=200,n(B)=300 and n(A∩B)=100. Then, n(A^c∩B^c )=

•  400
•  600
•  300
•  200

Solution
(c) We have, n(A^c∩B^c) =n{(A∪B)^c } =n(U)-n(A∪B) =n(U)-{n(A)+n(B)-n(A∩B)} =700-(200+300-100)=300

Q5. Consider the following statements: (i) Every reflexive relation is antisymmetric (ii) Every symmetric relation is antisymmetric Which one among (i) and (ii) is true?

•  (i) alone is true
•  (ii) alone is true
•  Both (i) and (ii) true
•   Neither (i) and (ii) is true

Solution
 D

Q6.  For real numbers x and y, we write x Ry⇔x-y+√2 is an irrational number. Then, the relation R is

•  Reflexive
•  Symmetric
• Transitive
•  None of these

Solution
(a) For any x∈R, we have x-x+√2=√2 an irrational number ⇒x R x for all x So, R is reflexive R is not symmetric, because √2 R 1 but 1 √2 R is not transitive also because √2 R 1 and 1 R 2 √2 but √2 2√2

Q7. If A,B and C are three non-empty sets such that A and B are disjoint and the number of elements contained in A is equal to those contained in the set of elements common to the sets A and C, then n(A∪B∪C) is necessarily equal to

•  n(B∪C)
•  n(A∪C)
•  Both (a) and (b)
•  None of these

Solution
(a) We have, A∩B=ϕ and A⊂C ⇒A∩B=ϕ and A∪C=C ∴n(A∪B∪C)=n(A∪C∪B)=n(C∪B)=n(B∪C)

Q8. If A={ϕ,{ϕ}}, then the power set of A is

•  A
•  T{ϕ,{ϕ},A}
•  {ϕ,{ϕ},{{ϕ}},A}
•  None of these

Solution
C

Q9. If R is a relation from a set A to a set B and S is a relation from B to a set C, then the relation SoR

•  Is from A to C
•  Is from C to A
•  Does not exist
•  None of these

Solution
(a)

Q10. If A_1⊂A_2⊂A_3⊂⋯⊂A_50 and n(A_i )=i-1, then n(⋂_(i=11)^50▒A_i )=

•  49
•  50
•  11
• 10

Solution
(d) It is given A_1⊂A_2⊂A_3⊂A_4…⊂A_50 ∴⋃_(i=11)^50▒〖A_i=A_11 〗 ⇒n(⋃_(i=11)^50▒A_i )=n(A_11 )=11-1=10