Advertisements
Advertisements
Question
Show that the relation R on the set Z of all integers, given by R = {(a,b) : 2 divides (a-b)} is an equivalence relation.
Advertisements
Solution
To prove relation is an equivalence relation
We have to show three properties
1. Reflexive
(a,a) ∈ R
2. Symmetric
(a,b) ∈ R
⇒ (b,a) ∈ R
3. Transitive
(a,b) ∈ R and (b,c) ∈ R
⇒ (a,c) ∈ R
1. R is reflexive because 2 divides (a - a)∀a ∈ z∀a ∈ z
2. 2 divides a - b
therefore, 2 divides b - a hence, (b,a) ∈ R
R is symmetric
3. (a, b) ∈ R
(b, c) ∈ R
then a − b and b − c are divisible by 2.
Now, a − c = ( a − b ) + ( b − c) = a−c
so, a − c is divisible by 2.
Therefore, (a, c ) ∈ R
Therefore, R is an equivalence relation.
APPEARS IN
RELATED QUESTIONS
Given an example of a relation. Which is reflexive and transitive but not symmetric.
Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?
Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is ______.
Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is ______.
Let R be a relation defined on the set of natural numbers N as
R = {(x, y) : x, y ∈ N, 2x + y = 41}
Find the domain and range of R. Also, verify whether R is (i) reflexive, (ii) symmetric (iii) transitive.
Let A = {a, b, c} and the relation R be defined on A as follows: R = {(a, a), (b, c), (a, b)}. Then, write minimum number of ordered pairs to be added in R to make it reflexive and transitive.
Defines a relation on N:
xy is square of an integer, x, y ∈ N
Determine the above relation is reflexive, symmetric and transitive.
If R and S are transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.
If R = {(x, y) : x2 + y2 ≤ 4; x, y ∈ Z} is a relation on Z, write the domain of R.
If R is a symmetric relation on a set A, then write a relation between R and R−1.
Let A = {3, 5, 7}, B = {2, 6, 10} and R be a relation from A to B defined by R = {(x, y) : x and y are relatively prime}. Then, write R and R−1.
State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} to be transitive ?
Let R be the equivalence relation on the set Z of the integers given by R = { (a, b) : 2 divides a - b }.
Write the equivalence class [0].
Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = a, b, c. Then, R is _______________ .
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6} Find (A × B) ∩ (A × C).
R = {(a, b) / b = a + 1, a ∈ Z, 0 < a < 5}. Find the Range of R.
Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is ______.
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 ______.
Every relation which is symmetric and transitive is also reflexive.
Let R be the relation on N defined as by x + 2 y = 8 The domain of R is ____________.
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 ____________.
Let A = {1, 2, 3, …. n} and B = {a, b}. Then the number of surjections from A into B is ____________.
Total number of equivalence relations defined in the set S = {a, b, c} is ____________.
A relation R in set A = {1, 2, 3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which of the following ordered pair in R shall be removed to make it an equivalence relation in A?
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever
Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:
R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}
- Mr. Shyam exercised his voting right in General Election-2019, then Mr. Shyam is related to which of the following?
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 relation R be defined by R = {(L1, L2): L1║L2 where L1, L2 ∈ L} then R is ____________ relation.
The relation > (greater than) on the set of real numbers is
In a group of 52 persons, 16 drink tea but not coffee, while 33 drink tea. How many persons drink coffee but not tea?
Define the relation R in the set N × N as follows:
For (a, b), (c, d) ∈ N × N, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in N × N.
