# Equivalence Class

## Trending Questions

**Q.**

What is equivalence class? Explain with example and vedio.

**Q.**Write the smallest equivalence relation on the set A = {1, 2, 3}.

**Q.**

What is the minimum number of elements an equivalence relation defined on the set A {1, 2, 3} would have?

**Q.**Let R={(P, Q) | P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1, −1) is the set :

- S={(x, y) | x2+y2=1}
- S={(x, y) | x2+y2=4}
- S={(x, y) | x2+y2=√2}
- S={(x, y) | x2+y2=2}

**Q.**

Show that each of the relation R in the set, given by

(i)

(ii)

is an equivalence relation. Find the set of all elements related to 1 in each case.

**Q.**Let A={x ∈Z:0≤x≤12}, R={(a, b):a, b∈A, |a−b| is divisible by 4}, then which among the following options are correct

- equivalence class of 1 is {1, 5, 9}
- R is reflexive and symmetric but not transitive relation.
- R is a Transitive relation
- equivalence class of 1 is A

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

(a) 1

(b) 2

(c) 3

(d) 4

**Q.**The maximum number of equivalence relations on the set A= { 1, 2, 3} are

- 1
- 2
- 4
- 5

**Q.**In how many different ways can a mixed doubles tennis game be organised between four married couples if no husband and wife play in the same game ?

**Q.**Number of equivalent in 44.8L of H2 at STP

**Q.**

In a class of 50 students, 30 students play cricket and 30 students play football and everyone plays at least one of these sports and no one plays any other sport. A relation is defined on the set of students such that aRb if ‘a’ and ‘b’ play a same sport. How many equivalence classes will be formed by this relation

**Q.**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].

[NCERT EXEMPLAR]

**Q.**

How many equivalence classes can be formed on a deck of cards, with respect to the relation "Belongs to the same suit"

13

52

4

26

**Q.**

Find :

x

**Q.**

**Q.**

What is the minimum number of elements an equivalence relation defined on the set A {1, 2, 3} would have?

**Q.**Let R={(P, Q) | P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1, −1) is the set :

- S={(x, y) | x2+y2=1}
- S={(x, y) | x2+y2=4}
- S={(x, y) | x2+y2=√2}
- S={(x, y) | x2+y2=2}

**Q.**

Show that the relation R in the set *A* = {1, 2, 3, 4, 5} given by

, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

**Q.**

Choose the correct answer in the following questions:

If is such that then

A.

B.

C.

D.

**Q.**

Write the first five terms of the following sequence and obtain the corresponding series:

**Q.**

How many of the following are not properties of equivalence classes?

- All elements of an equivalence class will be related to each other
- No element of an equivalence class will be related to an element of another equivalence class
- All the equivalence classes, which are sets, are disjoint
- Union of all the equivalence classes of particular relation will give the set A on which we defined the relation

**Q.**In the set Z of all integers, which of the following relation R is not an equivalence relation?

(a) x R y : if x ≤ y

(b) x R y : if x = y

(c) x R y : if x − y is an even integer

(d) x R y : if x ≡ y (mod 3)

**Q.**

On a set {a, b, c}, we define an equivalence relation 'R'. If this relation is {(a, a), (b, b), (c, c), (a, b), (b, a)}. How many equivalence classes will be formed from this relation?

1

0

2

3

**Q.**Let R be the equivalence relation on the set Z of integers given by R = {(a, b) : 2 divides a – b}. The equivalence class [0] is

- {0, 1, 2, 3, 4, 5, .....}
- {0, 2, 4, 6, 8, ......}
- {0, -2, -4, -6, -8, ......}
- (0, ±2, ±4, ±6, ±8, ......}

**Q.**The maximum number of equivalence relations on the set A= { 1, 2, 3} are

- 1
- 2
- 4
- 5

**Q.**If $A=\left[\begin{array}{c}3\\ 5\\ 2\end{array}\right]$ and B = [1 0 4], verify that (AB)

^{T}= B

^{T}A

^{T}

**Q.**Let R be the equivalence relation on the set Z of integers given by R = {(a, b) : 2 divides a – b}. The equivalence class [0] is

- {0, 1, 2, 3, 4, 5, .....}
- {0, 2, 4, 6, 8, ......}
- {0, -2, -4, -6, -8, ......}
- (0, ±2, ±4, ±6, ±8, ......}

**Q.**Let R be the equivalence relation on the set Z of integers given by R = {(a, b) : 2 divides a – b}. The equivalence class [0] is

- {0, 1, 2, 3, 4, 5, .....}
- {0, 2, 4, 6, 8, ......}
- {0, -2, -4, -6, -8, ......}
- (0, ±2, ±4, ±6, ±8, ......}

**Q.**Show that the relation R in the set A of points in a plane given by R={(P, Q):distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P≠(0, 0) is the circle passing through P with origin as centre.

**Q.**Write the value of the determinant $\left|\begin{array}{ccc}2& 3& 4\\ 5& 6& 8\\ 6x& 9x& 12x\end{array}\right|$