# Equivalence Relation

## Trending Questions

**Q.**Given the relation R={(1, 2), (2, 3)} on the set A={1, 2, 3}, the minimum number of ordered pairs required to make R an equivalence relation is

**Q.**

If $a+x=b+y=c+z+1$ where $a,b,c,x,y,z$are non-zero distinct real numbers, then $\left|\begin{array}{ccc}x& a+y& x+a\\ y& b+y& y+b\\ z& c+y& z+c\end{array}\right|$ is equals to

$y(a-b)$

$0$

$y(b-a)$

$y(a-c)$

**Q.**

The relation “is a subset of” on the power set $P\left(A\right)$ of a set $A$ is

Symmetric

Antisymmetric

Equivalency relation

None of these

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

- Reflexive and symmetric
- Transitive and symmetric
- Equivalence
- Reflexive, transitive but not symmetric

**Q.**

An integer $m$ is said to be related to another integer $n$ if $m$ is a multiple of $n$. Then the relation is

Reflexive & Symmetric

Reflexive and Transitive

Symmetric and Transitive

Equivalence Relation

**Q.**Let R be a relation from Q to Q defined by R={(a, b): a, b∈Q and a−b∈Z}.

Show that:

(i) (a, a)∈R for all a∈Q

(ii) (a, b)∈R implies that (b, a)∈R

(iii) (a, b)∈R and (b, c)∈R implies that (a, c)∈R

**Q.**

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a − b| is even}, 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.**

When a relation $R$ is an equivalence relation on a set $A$, then ${R}^{-1}$ is

Reflexive only

Symmetric but not transitive

An equivalence

None of the above

**Q.**

A relation “congruence modulo $\mathrm{m}$" is

Reflexive only

Transitive only

Symmetric only

An equivalence relation

**Q.**

If $R$ be a relation from $A=\{1,2,3,4\}$ to $B=\{1,3,5\}$ that is $(a,b)\in R$ if $a<b$, then $R\circ {R}^{-1}$ is

$\{(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)\}$

$\{(3,1),(5,1),(3,2),(5,2),(5,3),(5,4)\}$

$\{(3,3),(3,5),(5,3),(5,5)\}$

$\{(3,3),(3,4),(4,5)\}$

**Q.**

The total number of two digit numbers ‘$n$’, such that ${3}^{n}+{7}^{n}$ is a multiple of $10$, is

**Q.**On the set of positive rationals, a binary operation ∗ is defined by a∗b=2ab5. If 2∗x=3−1 then x=

- 25
- 16
- 12548
- 512

**Q.**Every relation which is symmetric and transitive is also reflexive.

- True
- False

**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.**If R is an equivalence relation on a set A, then R−1 is

- Reflexive relation only
- Symmetric but not transitive relation
- Reflexive, symmetric and transitive relation
- reflexive and transitive but not symmetric relation

**Q.**

Check
whether the relation R in R defined as R = {(*a*, *b*):
*a* ≤ *b*^{3}}
is reflexive, symmetric or transitive.

**Q.**

Let $A=\left\{p,q,r\right\}$. which of the following is an equivalence relation on $A$ ?

${R}_{1}=\left\{(p,q),(q,r),(p,r),(p,p)\right\}$

${R}_{2}=\left\{(r,p),(r,p),(r,r),(q,q)\right\}$

${R}_{3}=\left\{(p,p),(q,q),(r,r),(p,q)\right\}$

None of the above

**Q.**A relation R on the set of non-zero complex numbers defined by z1 R z2⟺z1−z2z1+z2 is real, then which of the following is not true?

- R is reflexive
- R is symmetric
- R is transitive
- R is not equivalance

**Q.**Let us define a relation R for real numbers as aRb if a ≥ b. Then R is

- an equivalence relation
- reflexive, transitive but not symmetric
- symmetric, transitive but not reflexive
- neither transitive nor reflexive but symmetric.

**Q.**

Given a non-empty set X. Consider P(X), which is the set of all subset of X. Defined the relation R in P(X) as follows:

For subsets A and B in P(X), ARB if and only if A⊂B. Is R an equivalence relation on P(X)? Justify your answer.

**Q.**

For any two real numbers $\theta $ and $\varphi $, we define $\theta R\varphi $, if and only if $se{c}^{2}\theta \u2013{\mathrm{tan}}^{2}\varphi =1$.

The relation $R$ is

reflexive but not transitive

symmetric but not reflexive

both reflexive and symmetric but not transitive

an equivalence relation

**Q.**

Let the positive numbers $a$, $b$, $c$, $d$ be in A.P. Then $abc$, $abd$, $acd$, $bcd$ are

NOT in A.P./G.P./H.P.

in A.P.

in G.P.

in H.P.

**Q.**Let A={2, 3, 4, 5....17, 18}. Let ≃ be the equivalence relation on A×A, cartesian product of A with itself, defined by (a, b)≃(c, d) if ad=bc. Then the number of ordered pairs of the equivalence class of (3, 2) is

**Q.**

Show that the relation R defined in the
set *A* of all triangles as R = {(*T*_{1}, *T*_{2}):
*T*_{1} is similar to *T*_{2}}, is
equivalence relation. Consider three right angle triangles *T*_{1}
with sides 3, 4, 5, *T*_{2} with sides 5, 12, 13 and *T*_{3}
with sides 6, 8, 10. Which triangles among *T*_{1}, *T*_{2}
and *T*_{3} are related?

**Q.**

Let A = {1, 2, 3, 4} and let R = {(2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then, R is

Equivalence relation

Symmetric

Transitive

Reflexive

**Q.**

The sum of$n$ terms of the series $\frac{4}{3}+\frac{10}{9}+\frac{28}{27}+.\dots $ is

**Q.**Let the relation ρ be defined on R by a ρ b holds if and only if a−b is zero or irrational then

- ρ is reflexive and transitive but is not symmetric
- ρ is reflexive only
- ρ is equivalence relation
- ρ is reflextive & symmetric but is not transitive

**Q.**

In a group of $15$people, $7$ read French, $8$ read English while $3$ of them read none of them. How many read French and English both?

$3$

$4$

$5$

$6$

**Q.**

If $\text{A}$ and $\text{B}$ are two equivalence relations defined on the set $\text{C}$, then

$\text{A}$ intersection $\text{B}$ is an equivalence relation

$\text{A}$ intersection $\text{B}$ is not an equivalence relation

$\text{AUB}$ is an equivalence relation

$\text{AUB}$ is not an equivalence relation

**Q.**On R, the relation ρ be defined by 'xρy holds if and only if x−y is zero or irrational". Then

- ρ is reflexive and transitive but not symmetric
- ρ is reflexive and symmetric but not transitive
- ρ is transitive and symmetric but not reflexive
- ρ is equivalence relation