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Sets, Relations and Functions

question 1

Let A = {15,16,17,18,0}, then the number of subsets of A containing 15 and 0 is

  1. 1
  2. 2
  3. 4
  4. 8
  5. none of these

question 2

In a group of 50 persons, everyone takes either tea or coffee. If 35 take tea and 25 take coffee, then the number of persons who take tea only is

  1. 10
  2. 25
  3. 35
  4. 40
  5. none of these

question 3

In a town of 10,000 families, it was found that 40% families buy newspaper A, 20% buy newspaper B, 10% buy newspaper C, 5% buy A & B, 3% buy B & C, and 4% buy A & C. If 2% families buy all the newspapers, then the number of families which buy none of A, B & C is

  1. 3300
  2. 1400
  3. 4000
  4. 1200
  5. none of these

question 4

If f : R→R is given by f(x) = 3x - 5, then f -1(x) is

  1. 3x - 5
  2. (3x - 5)-1
  3. (x + 5) / 3
  4. does not exist
  5. cannot be determined

question 5

The domain of the the real valued function Domain of real valued function is

  1. (1, 2)
  2. (-2, 2)
  3. (- ∞, - 2) ∪ (2, ∞)
  4. (- ∞, - 2) ∪ (1, ∞)
  5. (- ∞, ∞) - {1, ± 2}

question 6

The domain of the definition of the function Domain of a real valued function is

  1. (1, ∞)
  2. [1, ∞)
  3. (- ∞, - 1) ∪ (1, ∞)
  4. set of all reals different from 1
  5. none of these

question 7

The number of surjections from A = {1, 2, 3, ..., n}, n ≥ 2 onto B = {a, b} is

  1. nP2
  2. nC2
  3. 2n - 2
  4. 2n - 1
  5. none of these

question 8

Set A has 3 elements and set B has 4 elements. The number of injections that can be defined from A to B is

  1. 144
  2. 12
  3. 24
  4. 64
  5. none of these

question 9

If b2 - 4ac = 0 and a > 0, the domain of the function y = log {ax3 + (a + b)x2 + (b + c)x + c} is

  1. R
  2. R - {- b / 2a}
  3. (0, ∞)
  4. Domain of logarithmic function
  5. none of these

question 10

The range of the function Range of real valued function is

  1. R
  2. [3, ∞)
  3. (1/3, 3)
  4. [1/3, 3]
  5. none of these

syllabus

Sets – Introduction and Definition; Representation of sets, Roster method, Set –Builder method; Different types of sets – Empty set, Singleton set, Finite set, Infinite set, Equivalent sets, Equal sets; Cardinal number of a finite set; Subsets, Results on subsets; Power set; Universal set; Venn Diagrams, Representing sets using Venn Diagrams, Solving real world problems using Venn Diagram; Operations on sets - Union, Intersection, Difference, Symmetric difference and Complement of sets; Disjoint sets; Laws of Algebra of sets and proofs – Idempotent laws, Identity laws, Commutative laws, Associative laws, Distributive laws and De Morgan's laws; Results on Number of elements in sets.
Cartesian Product of sets, Ordered pair, Equality of ordered pairs, Graphical representation of Cartesian Product of sets; Results and Theorems on Cartesian Products; Relation, Total Number of Relations, Domain of a Relation, Range of a relation, Relation on a set, Inverse Relation; Types of Relations – Void Relation, Universal Relation, Identity Relation, Reflexive Relation, Symmetric Relation, Transitive Relation, Antisymmetric Relation and  Equivalence Relation; Prove Equivalence of a relation by checking Reflexivity, Symmetry and Transitivity; Results and Theorems on Relations.
Functions, Definition of Function; Domain, Co-domain and Range of a Function; Description or Representation of Function; Equal Functions; Function as a Relation; Standard Functions and their Graps – Constant Function, Identity Function, Modulus Function, Greatest Integer Function, Exponential Function, Logarithmic Function, Trigonometric Functions, Inverse Trigonometric Functions,, Hyperbolic Functions, Inverse Hyperbolic Functions; Kinds of Functions, One-One Function or Injection, Many-One Function, Onto Function or Surjection, Into Function, Bijection or One-One Onto Function; Proving Injectivity, Surjectivity and Bijectivity; Composition of Functions, Properties of Composition of Functions; Inverse of a Function, Properties of Inverse of a Function.