Equivalence Class

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?

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Q. Let $R=\left\{\right(P,Q\left)|P\text{and}Q\text{are at the same distance from the origin}\right\}$ be a relation, then the equivalence class of $(1,-1)$ is the set :

1. $S=\left\{(x,y)|x^2+y^2=1\right\}$
2. $S=\left\{(x,y)|x^2+y^2=4\right\}$
3. $S=\left\{(x,y)|x^2+y^2=\sqrt{2}\right\}$
4. $S=\left\{(x,y)|x^2+y^2=2\right\}$

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 $A=\{x$ $\in Z:0\le x\le 12\},$ $R=\left\{\right(a,b):a,b\in A,|a-b|\text{is divisible by}4\}$, then which among the following options are correct

1. equivalence class of $1$ is $\{1,5,9\}$
2. $R$ is reflexive and symmetric but not transitive relation.
3. $R$ is a Transitive relation
4. 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. 1
2. 2
3. 4
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

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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"

1. 13

2. 52

3. 4

4. 26

Q.

Find :

x

Q. (i) $R=\left\{\right(a,b):|a-b|\text{is a multiple of 4}\}$

Q.

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

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Q. Let $R=\left\{\right(P,Q\left)|P\text{and}Q\text{are at the same distance from the origin}\right\}$ be a relation, then the equivalence class of $(1,-1)$ is the set :

1. $S=\left\{(x,y)|x^2+y^2=1\right\}$
2. $S=\left\{(x,y)|x^2+y^2=4\right\}$
3. $S=\left\{(x,y)|x^2+y^2=\sqrt{2}\right\}$
4. $S=\left\{(x,y)|x^2+y^2=2\right\}$

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

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?

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

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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. 1

2. 0

3. 2

4. 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

1. {0, 1, 2, 3, 4, 5, .....}
2. {0, 2, 4, 6, 8, ......}
3. {0, -2, -4, -6, -8, ......}
4. $\left(0,\pm 2,\pm 4,\pm 6,\pm 8,......\right\}$

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

1. 1
2. 2
3. 4
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

1. {0, 1, 2, 3, 4, 5, .....}
2. {0, 2, 4, 6, 8, ......}
3. {0, -2, -4, -6, -8, ......}
4. $\left(0,\pm 2,\pm 4,\pm 6,\pm 8,......\right\}$

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

1. {0, 1, 2, 3, 4, 5, .....}
2. {0, 2, 4, 6, 8, ......}
3. {0, -2, -4, -6, -8, ......}
4. $\left(0,\pm 2,\pm 4,\pm 6,\pm 8,......\right\}$

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

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