Relations and Functions - Online Test
Q1. If A is a finite set containing n distinct elements, then the number of relations on A is equal to
Answer : Option D
Explaination / Solution:
The number of elements in A x A is n x n = n2. hence ,the number of relations on A = number of subsets of A x A = 2nxn =.
Q2. Let A = {1, 2, 3}, then the domain of the relation R = {(1, 1), (2, 3), (2, 1)} defined on A is
Answer : Option A
Explaination / Solution:
Since the domain is represented by the x- co ordinate of the ordered pair ( x , y ).Therefore, domain of the given relation is { 1 , 2 }.
Q3. Let A = {a, b, c} then the range of the relation R = {(a, b), (a, c), (b, c)} defined on A is
Answer : Option B
Explaination / Solution:
Since the range is represented by the y- co ordinate of the ordered pair ( x , y ).Therefore, range of the given relation is { b , c }.
Q4. Number of relations that can be defined on the set A = {a, b, c, d} is
Answer : Option A
Explaination / Solution:
No. of elements in the set A = 4 .Therefore , the no. of elements in As, the no. of relations in = no. of subsets of .
Q5. Let A = {1, 2, 3, 4, 5, 6}. Which of the following partitions of A correspond to an equivalence relation on A?
Answer : Option B
Explaination / Solution:
Conditions for the partition sub-sets to be an equivalence relation:
(i) The partition sub-sets must be disjoint i.e.their is no common elements between them
(ii) Their union must be equal to the main set (super-set)
Here the set A={1,2,3,4,5,6},the partition sub-sets {1,3},{2,4,5},{6} are pairwise disjoint and their union i.e. {1,3} U {2,4,5} U {6} = {1,2,3,4,5,6} = A,which is the condition for the partition sub-sets to be an equivalence relation of the set A.
Q6. A relation R on a non – empty set A is an equivalence relation iff it is
Answer : Option B
Explaination / Solution:
By definition of Equivalence Relation,a relation is said to be equivalence if it is reflexive,symmetric and transitive
Q7.
Let and x, y R} be a relation in R. The relation R is
Answer : Option B
Explaination / Solution:
A relation R on a non empty set A is said to be symmetric if fx RyyRx, for all x , y R Clearly is same as for all x , y R. Therefore ,R is symmetric.
Q8. The void relation ( a subset of A x A ) on a non empty set A is :
Answer : Option C
Explaination / Solution:
The relation { }⊂ A x A on a is surely not reflexive.However ,neither symmetry nor transitivity is contradicted .So { } is a transitive and symmetric relation on A.
Q9. f n ⩾ 2 , then the number of onto mappings or surjections that can be defined from { 1,2,3,4,………..,n} onto {1,2} is
Answer : Option A
Explaination / Solution:
Explanation:
The number of onto functions that can be defined from a finite set A containing n elements onto a finite set B containing 2 elements == .
Q10. If A = { 1, 2, 3}, then the relation R = {(1, 2), (2, 3), (1, 3)} in A is
Answer : Option D
Explaination / Solution:
A relation R on a non empty set A is said to be transitive if xRy and y Rz⇒xRz, for all x ∈ R. Here , (1, 2) and (2, 3) belongs to R implies that (1, 3) belongs to R.
