Relations And Functions Test

Relations And Functions

This is Relations and functions Test-10 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
Let T be the set of all triangles in the Euclidean plane, and let a relation R on The defined as a Rb if a is congruent to b a,\(b \in T\) . Then R is
  • A
    reflexive but not transitive
  • B
    transitive but not symmetric
  • C
    equivalence
    Correct
  • D
    none of these
2
Consider the non – empty set consisting of children in a family and a relation Rdefined as aRb if a is brother of b. Then R is
  • A
    both symmetric and transitive
  • B
    neither symmetric nor transitive
  • C
    symmetric but not transitive
  • D
    transitive but not symmetric
    Correct
3
The maximum number of equivalence relations on the set A = {1, 2, 3} are
  • A
    5
    Correct
  • B
    1
  • C
    3
  • D
    2
4
If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is
  • A
    symmetric
  • B
    reflexive
  • C
    transitive
    Correct
  • D
    none of these
5
Let us define a relation R in R as aRb if \(a{\text{ }} \geqslant {\text{ }}b\) . Then R is
  • A
    reflexive, transitive but notsymmetric
    Correct
  • B
    neither transitive nor reflexive
  • C
    symmetric, transitive butnot reflexive
  • D
    an equivalence relation
6
Let A = {1, 2, 3} and consider the relation R = {1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}. Then R is
  • A
    symmetric and transitive
  • B
    reflexive but not transitive
  • C
    neither symmetric, nor transitive
  • D
    As (1, 1), (2, 2), (3, 3)\( \in R\) , therefore R is reflexive. Since (1,2) \( \in R\;, \) but (2,1) R therefore R is not symmetric. reflexive but not symmetric
    Correct
7
The identity element for the binary operation * defined on \( Q\sim \left \{ 0 \right \} \) as a * b = a, b ∈ \( Q\sim \left \{ 0 \right \} \) is
  • A
    2
    Correct
  • B
    none of these
  • C
    1
  • D
    0
8
Given an arbitrary equivalencerelation R in an arbitrary set X, R divides X into
  • A
    three sets
  • B
    mutually disjoint subsets
    Correct
  • C
    two sets
  • D
    intersecting sets
9
A binary operation ∗ : \(:{\text{ }}A{\text{ }} \times {\text{ }}A{\text{ }} \to {\text{ }}A\) is said to be associative if
  • A
    Option A
  • B
    Option B
  • C
    Option C
  • D
    Option D
    Correct
10
Equivalence classes are
  • A
    mutually disjoint subsets
    Correct
  • B
    trivial sets
  • C
    intersecting sets
  • D
    power sets
11
Equivalence classes \({A_i}\) satisfy
  • A
    some elements of \({A_i}\) are related to each other, for all i.
  • B
    all elements of \({A_i}\) are related to each other, for all i.
    Correct
  • C
    all elements of \({A_i}\) are not related to each other, for all i.
  • D
    all elements of \({A_i}\) are related to each other, for somei.
12
Equivalence classes \({A_i}\) satisfy
  • A
    all elements of \({A_i}\) are related to any element of \({A_j},{\text{ }}i{\text{ }} \ne {\text{ }}j\)
  • B
    no element of \({A_i}\) is related to any element of \({A_j},{\text{ }}i{\text{ }} \ne {\text{ }}j\)
    Correct
  • C
    no element of \({A_i}\) is related to any element of \({A_i}\)
  • D
    some elements of \({A_i}\) are related to any element of \({A_j},{\text{ }}i{\text{ }} \ne {\text{ }}j\)
13
Equivalence classes \({A_i}\) satisfy
  • A
    \( \cup {A_j} = {\text{ }}X{\text{ }}and{\text{ }}Ai{\text{ }} \cap {\text{ }}{A_j} \) \(= {\text{ }}\varphi ,{\text{ }}i{\text{ }} \ne {\text{ }}j.\)
    Correct
  • B
    \( \cup {A_j} \ne {\text{ }}X{\text{ }}\)and\({\text{ }}Ai{\text{ }} \cap {\text{ }}{A_j} \ne {\text{ }}\varphi ,{\text{ }}i{\text{ }} \ne {\text{ }}j.\)
  • C
    \( \cup {A_j} = {\text{ }}X{\text{ }}\) and \({\text{ }}Ai{\text{ }} \cap {\text{ }}{A_j} \ne {\text{ }}\varphi ,{\text{ }}i{\text{ }} \ne {\text{ }}j.\)
  • D
    \( \cup {A_j} \ne {\text{ }}X{\text{ }}\)and \({\text{ }}Ai{\text{ }} \cap {\text{ }}{A_j} = {\text{ }}\varphi ,{\text{ }}i{\text{ }} \ne {\text{ }}j.\)
14
Pick the true statement from the following
  • A
    Every function is invertible.
  • B
    A binary operation on a set has always the identity element.
  • C
    The composition of functions is commutative.
  • D
    The composition of functions is associative.
    Correct
15
Let A = { 2 , 3 , 6 }. Which of the following relations on A are reflexive ?
  • A
    \({R_2}\) = { ( 2,2 ) , ( 3,3 ) , ( 3,6 ) , ( 6,3 ) }
  • B
    none of these
  • C
    \({R_1}\) = { ( 2,2 ) , ( 3,3 ) , ( 6,6 ) }
    Correct
  • D
    \({R_3}\) = { ( 2,2 ) , ( 3,6 ) , ( 2,6 ) }