# Equivalence Relation

## Trending Questions

**Q.**

Let $n$ be a fixed positive integer. Define a relation $R$ on the set $Z$ of integers by, $aRb\iff n|\left(a\u2013b\right)$. Then $R$ is:

Reflexive

Symmetric

Transitive

Equivalence

All the above

**Q.**Let W denote the words in the English dictionary. Define the relation R by R={(x, y)ϵW×W| the words x and y have at least one letter in common.} Then R is

- relexive, symmetric and not transitive
- relexive, symmetric and transitive
- reflexive, not symmetric and transitive
- not reflexive, symmetric and transitive

**Q.**Let U be the set of all boys and girls in a school, G be the set of all girls in the school, B be the set of all boys in the school, and S be the set of all students in the school who take swimming. Some, but not all, students in the school take swimming. Draw a Venn diagram showing one of the possible interrelationship among sets U, G, B and S. 12. For all sets A, B and C, show that (A – B) n (C – B) = A – (B . C)

**Q.**what is alpha , beta , gamma and theta?

**Q.**If n(A) = 3 and n(B) = 2 then number of relations from A to B is ____

- 9
- 8
- 64
- 6

**Q.**

$R\subseteq A\times A,(A\ne 0)$ is an equivalence relation if $R$ is

Reflexive, symmetric but not transitive

Reflexive, neither symmetric nor transitive

Reflexive, symmetric and transitive

None of the above

**Q.**Let R be a relation on the set N be defined by {(x, y)|x, yϵN, 2x+y=41}. Then R is

- Reflexive
- Symmetric
- Transitive
- None of these

**Q.**Consider the following relations:

R = {(x, y) | x, y are real numbers and x = wy for some rational number w};

S = {(mn, pq)| m, n, p and q are integers such that n, q ≠0 and qm = pn}.

Then

- R and S both are equivalence relations
- R is an equivalence relation but S is not an equivalence relation
- Neither R nor S is an equivalence relation
- S is an equivalence relation but R is not an equivalence relation.

**Q.**

Consider the relations $R=\{(x,y)|x,y$ are real numbers and $x=wy$ for some rational number $w\}$ $S=\{(m/n,p/q)|m,n,p$ and $q$ are integers such that $n,q$ not equal to $0$ and $qm=pn\}$. Then

$R$ is an equivalence relation but $S$ is not an equivalence relation

Neither $R$ nor $S$ is an equivalence relation

$S$ is an equivalence relation but $R$ is not an equivalence relation

$R$ and $S$ both are equivalence relations

**Q.**Let R be a relation defined in the set of real numbers by a R b⇔1+ab>0. Then R is

- Equivalence relation
- Transitive
- Symmetric
- Anti-symmetric

**Q.**Let R be a relation over the set N×n and it is defined by (a, b) R (c, d) ⇒ a+ d = b + c. Then, R is

- reflexive only
- symmetric only
- an equivalence relation
- transitive only

**Q.**The relation R defined on the set N of natural numbers by xRy⇔2x2−3xy+y2=0 is

- symemtric but not reflexive
- only symmetric
- not symmetric but reflexive
- Reflexive and symmetric

**Q.**Let A = {1, 2, 3, 4} and R be a relation in A given by R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1), (1, 3)}. Then R is

- symmetric
- Reflexive
- transitive
- an equivalence relation

**Q.**Let R be a relation defined on the set of natural numbers N as

R={(x, y):xϵN, yϵN, 2x+y=41} Check if

R is (i) reflexive (ii) symmetric

**Q.**

Is congruent to" on the set of all triangles is an equivalence relation

True

False

**Q.**Let R be a relation on the set N defined by {(x, y)ϵN, 2x+y=41}. Then R is

- reflexive
- transitive
- symmetric
- none of these

**Q.**

Let R be an equivalence relation on a finite set A having n elements. Then the number of ordered pairs in R is

Less than

*n*Greater than or equal to

*n*Less than or equal to

*n*None of these

**Q.**Let S be the set of all real numbers and let R be a relation on S, defined by a R b ⇔ |a-b| ≤ 1. Then, R is

- reflexive and transitive but not symmetric
- reflexive and symmetric but not transitive
- symmetric and transitive but not reflexive
- an equivalence relation.

**Q.**Two points A and B in a plane are related if OA=OB, where O is a fixed point. This relation is

- equivalence relation
- partial order relation
- reflexive but not symmetric
- refleve but not transitive.

**Q.**Consider the following relations:

R = {(x, y) | x, y are real numbers and x = wy for some rational number w};

S = {(mn, pq)| m, n, p and q are integers such that n, q ≠0 and qm = pn}.

Then

- S is an equivalence relation but R is not an equivalence relation.
- R and S both are equivalence relations
- R is an equivalence relation but S is not an equivalence relation
- Neither R nor S is an equivalence relation

**Q.**If we define a relation R on the set N×N as (a, b)R(c, d)⇔a+d=b+c for all (a, b), (c, d)ϵN×N, then the relation is

- symmetric and transitive only
- equivalence relation
- symmetric only
- reflexive only

**Q.**

On a set $N$ of all natural numbers is defined the relation $R$ by $aRb$ iff the $GCD$ of $a$ and $b$ is $2$, then $R$ is

Reflexive and Transitive

Symmetric and Transitive

Symmetric only

Not reflexive, not symmetric, not transitive

**Q.**let x={1, 2, 3, 4, 5}.the no. of different ordered pairs (y, z) that can be formed such that y is subset of x, z is subset of x and y intersection z is empty is

**Q.**Which of the following relations in R is an equivalence relatilon?

- xR1 y⇔|x|=|y|
- xR3 y⇔xy
- xR2 y⇔x≥y
- xR4 y⇔x<y

**Q.**Let R be a relation defined by R = (a, b):a≥b, a, bϵR. The relation R is

- Reflexive, symmetric and transitive
- Reflexive, transitive but not symmetric
- Neither transitive nor reflexive but symmetric
- Symmetric, transitive but not reflexive

**Q.**Choose best suited word for the following example. Let A be a family of sets and let R be the relation in A defined by 'x' is a subset of 'y'. So R is-

- A set
- An equivalence relation
- A relation
- None of the above

**Q.**

The empty relation defined on a set of real numbers is transitive.

Flase

True

**Q.**Which of the following relations in R is an equivalence relatilon?

- xR1 y⇔|x|=|y|
- xR2 y⇔x≥y
- xR4 y⇔x<y
- xR3 y⇔xy

**Q.**The relation R defined on the set N of natural numbers by xRy⇔2x2−3xy+y2=0 is

- not symmetric but reflexive
- Reflexive and symmetric
- symemtric but not reflexive
- only symmetric