मराठी

Define an Equivalence Relation ?

Advertisements
Advertisements

प्रश्न

Define an equivalence relation ?

व्याख्या
Advertisements

उत्तर

A relation R on set A is said to be an equivalence relation if
(i) it is reflexive,
(ii) it is symmetric and
(iii) it is transitive.

Relation R on set A satisfying all the above three properties is an equivalence relation.

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 1: Relations - Exercise 1.3 [पृष्ठ ३०]

APPEARS IN

आर.डी. शर्मा Mathematics Volume 1 and 2 [English] Class 12
पाठ 1 Relations
Exercise 1.3 | Q 13 | पृष्ठ ३०

संबंधित प्रश्‍न

Determine whether the following relation is reflexive, symmetric and transitive:

Relation R in the set Z of all integers defined as R = {(x, y) : x – y is an integer}.


Given an example of a relation. Which is symmetric and transitive but not reflexive.


Let L be the set of all lines in the XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.


Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is ______.


The binary operation *: R x R → R is defined as a *b = 2a + b Find (2 * 3)*4


Test whether the following relation R1 is  (i) reflexive (ii) symmetric and (iii) transitive :

R1 on Q0 defined by (a, b) ∈ R1 ⇔ = 1/b.


Test whether the following relation R2 is (i) reflexive (ii) symmetric and (iii) transitive:

R2 on Z defined by (a, b) ∈ R2 ⇔ |a – b| ≤ 5


Test whether the following relation R3 is (i) reflexive (ii) symmetric and (iii) transitive:

R3 on R is defined by (a, b) ∈ R3 `⇔` a2 – 4ab + 3b2 = 0.


Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R2 = {(2, 2), (3, 1), (1, 3)}, R3 = {(1, 3), (3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is (i) reflexive (ii) symmetric (iii) transitive.


Let R be a relation defined on the set of natural numbers N as
R = {(xy) : x N, 2x + y = 41}
Find the domain and range of R. Also, verify whether R is (i) reflexive, (ii) symmetric (iii) transitive.


Defines a relation on N:

x + 4y = 10, x, y ∈ N

Determine the above relation is reflexive, symmetric and transitive.


Show that the relation R on the set Z of integers, given by
R = {(a, b) : 2 divides a – b},  is an equivalence relation.


Let S be a relation on the set R of all real numbers defined by
S = {(a, b) ∈ R × R : a2 + b2 = 1}
Prove that S is not an equivalence relation on R.


Define a reflexive relation ?


A = {1, 2, 3, 4, 5, 6, 7, 8} and if R = {(xy) : y is one half of xxy ∈ A} is a relation on A, then write R as a set of ordered pairs.


Let A = {0, 1, 2, 3} and R be a relation on A defined as
R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}
Is R reflexive? symmetric? transitive?


Write the smallest equivalence relation on the set A = {1, 2, 3} ?


Mark the correct alternative in the following question:

Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then, R is _____________ .


Mark the correct alternative in the following question:

Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if l is perpendicular to m for all l, m  L. Then, R is ______________ .


Mark the correct alternative in the following question:

Consider a non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then, R is _____________ .


If A = {1, 2, 3, 4 }, define relations on A which have properties of being: 
reflexive, symmetric and transitive


Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
a mapping from A to B which is not injective


Give an example of a map which is neither one-one nor onto


The following defines a relation on N:
x + y = 10, x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.


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 ______.


The relation R on the set A = {1, 2, 3} defined as R = {{1, 1), (1, 2), (2, 1), (3, 3)} is reflexive, symmetric and transitive.


Which of the following is not an equivalence relation on I, the set of integers: x, y


Let A = {1, 2, 3}, then the domain of the relation R = {(1, 1), (2, 3), (2, 1)} defined on A is ____________.


Let `"f"("x") = ("x" - 1)/("x" + 1),` then f(f(x)) is ____________.


Let S = {1, 2, 3, 4, 5} and let A = S x S. Define the relation R on A as follows:
(a, b) R (c, d) iff ad = cb. Then, R is ____________.


Given set A = {1, 2, 3} and a relation R = {(1, 2), (2, 1)}, the relation R will be ____________.


Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

Answer the following using the above information.

  • Let R = {(L1, L2 ): L1 is parallel to L2 and L1: y = x – 4} then which of the following can be taken as L2?

The value of k for which the system of equations x + ky + 3z = 0, 4x + 3y + kz = 0, 2x + y + 2z = 0 has nontrivial solution is


The number of surjective functions from A to B where A = {1, 2, 3, 4} and B = {a, b} is


A relation 'R' in a set 'A' is called a universal relation, if each element of' A' is related to :-


If f(x + 2a) = f(x – 2a), then f(x) is:


Let R1 and R2 be two relations defined as follows :

R1 = {(a, b) ∈ R2 : a2 + b2 ∈ Q} and

R2 = {(a, b) ∈ R2 : a2 + b2 ∉ Q}, where Q is the set of all rational numbers. Then ______


Which relation is reflexive on \(A=\{1,2\}\)?


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×