Equivalence Relation

Q. Given the relation $R=\left\{\right(1,2),(2,3\left)\right\}$ 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

1. $y(a-b)$

2. $0$

3. $y(b-a)$

4. $y(a-c)$

Q.

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

1. Symmetric

2. Antisymmetric

3. Equivalency relation

4. 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

1. Reflexive and symmetric
2. Transitive and symmetric
3. Equivalence
4. 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

1. Reflexive & Symmetric

2. Reflexive and Transitive

3. Symmetric and Transitive

4. Equivalence Relation

Q. Let $R$ be a relation from $Q$ to $Q$ defined by $R=\left\{\right(a,b):a,b\in Q\text{and}a-b\in Z\}$.
Show that:
$\left(i\right)(a,a)\in R$ for all $a\in Q$
$\left(ii\right)(a,b)\in R$ implies that $(b,a)\in R$
$\left(iii\right)(a,b)\in R\text{and}(b,c)\in R$ implies that $(a,c)\in R$

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

1. Reflexive only

2. Symmetric but not transitive

3. An equivalence

4. None of the above

Q.

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

1. Reflexive only

2. Transitive only

3. Symmetric only

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

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

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

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

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 $\ast$ is defined by $a\ast b=\frac{2ab}{5}.$ If $2\ast x=3^{-1}$ then $x=$

1. $\frac{2}{5}$
2. $\frac{1}{6}$
3. $\frac{125}{48}$
4. $\frac{5}{12}$

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

1. True
2. 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

1. Reflexive relation only
2. Symmetric but not transitive relation
3. Reflexive, symmetric and transitive relation
4. reflexive and transitive but not symmetric relation

Q.

Check whether the relation R in R defined as R = {(a, b): a ≤ b³} is reflexive, symmetric or transitive.

Q.

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

1. $R_1=\left\{(p,q),(q,r),(p,r),(p,p)\right\}$

2. $R_2=\left\{(r,p),(r,p),(r,r),(q,q)\right\}$

3. $R_3=\left\{(p,p),(q,q),(r,r),(p,q)\right\}$

4. None of the above

Q. A relation $R$ on the set of non-zero complex numbers defined by $z_{1}R{z}_2⟺\frac{z_1-z_2}{z_1+z_2}$ is real, then which of the following is not true?

1. $R$ is reflexive
2. $R$ is symmetric
3. $R$ is transitive
4. $R$ is not equivalance

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

1. an equivalence relation
2. reflexive, transitive but not symmetric
3. symmetric, transitive but not reflexive
4. 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 $A\subset B$. 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 $se{c}^2\theta –{\tan }^2\varphi =1$.

The relation $R$ is

1. reflexive but not transitive

2. symmetric but not reflexive

3. both reflexive and symmetric but not transitive

4. an equivalence relation

Q.

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

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

2. in A.P.

3. in G.P.

4. in H.P.

Q. Let $A=\{2,3,4,\mathrm{5....17},18\}.$ Let $\simeq$ be the equivalence relation on $A\times A,$ cartesian product of $A$ with itself, defined by $(a,b)\simeq (c,d)$ 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 = {(T₁, T₂): T₁ is similar to T₂}, is equivalence relation. Consider three right angle triangles T₁ with sides 3, 4, 5, T₂ with sides 5, 12, 13 and T₃ with sides 6, 8, 10. Which triangles among T₁, T₂ and T₃ 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

1. Equivalence relation

2. Symmetric

3. Transitive

4. 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 $\rho$ be defined on $R$ by $a\rho b$ holds if and only if $a-b$ is zero or irrational then

1. $\rho$ is reflexive and transitive but is not symmetric
2. $\rho$ is reflexive only
3. $\rho$ is equivalence relation
4. $\rho$ 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?

1. $3$

2. $4$

3. $5$

4. $6$

Q.

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

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

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

3. $\text{A U B}$ is an equivalence relation

4. $\text{A U B}$ is not an equivalence relation

Q. On $R$, the relation $\rho$ be defined by '$x\rho y$ holds if and only if $x-y$ is zero or irrational". Then

1. $\rho$ is reflexive and transitive but not symmetric
2. $\rho$ is reflexive and symmetric but not transitive
3. $\rho$ is transitive and symmetric but not reflexive
4. $\rho$ is equivalence relation