SETS-1 Quiz-4

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. 31. In Q.No. 6,⋂_(n=3)^10▒〖A_n=〗
  •  {3,5,7,11,13,17,19}
  •  {2,3,5}
  •  {2,3,5,7,11,13,17}
  •  {3,5,7}
(b) We have, A_2⊂A_3⊂A_4⊂⋯⊂A_10 ∴⋂_(n=3)^10▒〖A_n=A_3={2,3,5}〗

Q2.If A={x∶x is a multiple of 3} and, B={x∶x is a multiple of 5}, then A-B is
  •  A ̅∩B
  •  A∩B ̅
  •  A ̅∩B ̅
  •  (A∩B) ̅
(b) Clearly, x∈A-B⇒x∈A but x∉B ⇒x is a multiple of 3 but it is not a multiple of 5 ⇒x∈A∩B ̅

Q3.  If A is a non-empty set, then which of the following is false? p∶ Every reflexive relation is a symmetric relation q∶ Every antisymmetric relation is reflexive Which of the following is/are true?
  •   p alone
  •  q alone
  •  Both p and q
  •  Neither p nor q
(d) If A={1,2,3}, then R={(1,1),(2,2),(3,3),(1,2)} is reflexive on A but it is not symmetric So, a reflexive relation need not be symmetric The relation ‘is less than’ on the set Z of integers is antisymmetric but it is not reflexive

Q4. Let A be a set represented by the squares of natural number and x,y are any two elements of A. Then,
  •  x-y∈A
  •  xy∈A
  •  x+y∈A
  •  x/y∈A
(b) Let x,y∈A. Then, x=m^2,y=n^2 for some m,n∈N ⇒xy=(mn)^2∈A

Q5. Let R be a reflexive relation on a set A and I be the identity relation on A. Then,
  •  R⊂I
  •  I⊂R
  •  R = I
  •  I⊂R

Q6.  36. If A and B are two sets such that n(A)=7,n(B)=6 and (A∩B)≠ϕ. Then the greatest possible value of n(A Δ B), is
  •  11
  •  12
  • 13
  •  10
(a) We have, A ∆ B=(A∪B)-(A∪B) ⇒n(A ∆ B)=n(A)+n(B)-2 n(A∩B) So, n(A ∆ B) is greatest when n(A∩B) is least It is given that A∩B≠ϕ. So, least number of elements in A ∩B can be one ∴ Greatest possible value of n(A ∆ B) is 7+6-2×1=11

Q7. A class has 175 students. The following data shows the number of students opting one or more subjects. Mathematics 100; Physics 70; Chemistry 40; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. Hoe many students have offered Mathematics alone?
  •  38
  •  48
  •  60
  •  22
(c) Given, n(M)=100,n(P)=70, n(C)=40 n(M∩P)=30, n(M∩C)=28, n(P∩C)=23 and n(M∩P∩C)=18 ∴n(M∩P^'∩C^' )=n[M∩(P∩C')] =n(M)-n[M∩(P∩C)] =n(M)-[n(M∩P)+n(M∩C)-n(M∩P∩C)] =100-[30+28-18=60]

Q8. Let U be the universal set for sets A and B such that n(A)=200,n(B)=300 and n(A∩B)=100. Then, n(A'∩B') is equal to 300, provided that n( U) is equal to
  •  600
  •  T700
  •  800
  •  900
(b) We have, n(A^'∩B^' )=n((A∪B)') ⇒n(A^'∩B^' )=n(U)-n(A∪B) ⇒n(A^'∩B^' )=n(U)-{n(A)+n(B)-n(A∩B)} ⇒300=n (U)-{200+300-100} ⇒n(U)=700

Q9. In an office, every employee likes at least one of tea, coffee and milk. The number of employees who like only tea, only coffee, only milk and all the three are all equal. The number of employees who like only tea and coffee, only coffee and milk and only tea and milk are equal and each is equal to the number of employees who like all the three. Then a possible value of the number of employees in the office is
  •  65
  •  90
  •  77
  •  85
(c) Total number of employees =7x i.e. a multiple of 7. Hence, option (c) is correct

Q10. If A={x,y,z}, then the relation R={(x,x),(y,y),(z,z),(z,x),(z,y)} is
  •  Symmetric
  •  Antisymmetric
  •  Transitive
  • Both (a) and (b)
(d) Clearly, R is an equivalence relation on A

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