Reflexive Relations

Q. Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by
$R=\left\{\right(x,y)\in N\times N:x^3-3x^{2}y-x{y}^2+3y^3=0\}$.
Then the relation $R$ is

1. an equivalence relation
2. reflexive and symmetric, but not transitive
3. reflexive but neither symmetric nor transitive
4. symmetric but neither reflexive nor transitive

Q. If $A=\{1,2,3\},$ then number of reflexive relations that can be defined on $A$ is

Q. Which of the following is not correct for relation $R$ on the set of real numbers?

1. $(x,y)\in R\iff |x–y|\le 1$ is reflexive and symmetric
2. $(x,y)\in R\iff 0<|x|–|y|\le 1$ is neither transitive nor symmetric
3. $(x,y)\in R\iff 0<|x–y|\le 1$ is symmetric and transitive
4. $(x,y)\in R\iff |x|–|y|\le 1$ is reflexive but not symmetric

Q.

What is the difference between Identity relation and reflexive relation ?

Q.

Let $R$ be relation on the set $N$ be defined by $\left\{(x,y)|x,y\in N,2x+y=41\right\}$. Then $R$ is

1. Reflexive

2. Symmetric

3. Transitive

4. None of these

Q.

Let $R$ be a reflexive relation on a set $A$ and $I$ be the identity relation on $A$. Then:

1. $R\subset I$

2. $I\subset R$

3. $R=1$

4. None of these

Q.

Let $R_1$ be a relation defined by $R_1=\left\{\left(a,b\right)|a\ge b,a,b\in \mathrm{ℝ}\right\}$. Then $R_1$ is

1. Reflexive, transitive and symmetric

2. Reflexive, transitive but not symmetric

3. Symmetric, transitive but not reflexive

4. Neither transitive nor reflexive but symmetric

Q.

In the set $A=\left\{1,2,3,4,5\right\}$ a relation $R$ is defined by $R=\left\{(x,y)|x,y\in Aandx<y\right\}$. Then $R$ is

1. Reflexive

2. Symmetric

3. Transitive

4. None of these

Q.

Cheeck whether the relation R. defined in the set {1, 2, 3, 4, 5, 6} as R ={(a, b):b=a+1} is reflexive, symmetric or transitive.

Q.

The relation R defined on the set of natural numbers as $\left\{(a,b):a\mathrm{differs}\mathrm{from}b\mathrm{by}3\right\}$ is given by:

1. $\left\{(1,4),(2,5),(3,6)...\right\}$

2. $\left\{(4,1),(5,2),(6,3),...\right\}$

3. $\left\{(1,3),(2,6),(1,3),...\right\}$

4. None of the above

Q. For the set A = {1, 2, 3}, define a relation R on the set A as follows:
R = {(1, 1), (2, 2), (3, 3), (1, 3)}
Write the ordered pairs to be added to R to make the smallest equivalence relation.

Q.

$x^2=xy$ is a relation which is

1. Symmetric

2. Reflexive

3. Transitive

4. All of these

Q.

Let $R=\left\{(1,3),(4,2),(2,4),(2,3),(3,1)\right\}$ be relation on the set $A=\{1,2,3,4\}$. The relation $R$ is

1. Reflexive

2. Transitive

3. Not symmetric

4. A function

Q. If a relation $R$ defined on set of real numbers as $xRy⟺x-y+\sqrt{2}$ is an irrational number. Then the relation $R$ is

1. a reflexive relation
2. a symmetric relation
3. a transitive relation
4. both reflexive and transitive relation

Q.

Given an example of a relation. Which is
(v) Symmetric and transitive but not reflexive.

Q.

Let $\mathrm{S}$ be the set of all real numbers. A relation $\mathrm{R}$ has been defined on $\mathrm{S}$ by $a\mathrm{Rb}=|a-b|\le 1$, then $R$ is.

1. Symmetric and transitive but not reflexive

2. Reflexive and transitive but not symmetric

3. Reflexive and symmetric but not transitive

4. an equivalence relation

Q.

Let A ={1, 2, 3}.Then, number of equivalence relations containing (1, 2) is
(a)1
(b)2
(c)3
(d)4

Q. Let $R$ be a reflexive relation on a set $A$ and $I$ be the identity relation on $A.$ Then

1. $R\subseteq I$
2. $I\subset R$
3. $R\subset I$
4. $I\subseteq R$

Q. The number of reflexive relations on a set with four elements is equal to

1. 2048
2. 4096
3. 256
4. 8192

Q.

Universal relation defined on a set A is reflexive.

1. True

2. False

Q.

If $r$ is a relation from $R$ (set of real numbers) to $R$ defined by $r=\left\{(a,b)|a,b\in R;a-b+\sqrt{3}\text{is an irrational number}\right\}$. Then, the relation r is

1. an equivalence relation

2. only reflexive

3. only symmetric

4. only transitive

Q. On the set $R$ of real numbers we define $xPy$ if and only if $xy\ge 0$. Then the relation $P$ is

1. reflexive but not symmetric
2. symmetric but not reflexive
3. transitive but not reflexive
4. reflexive and symmetric but not transitive

Q. If $n\left(A\right)=m,m>0$, then number of reflexive relations from $A$ to $A$ is

1. $2^{m^2}$
2. $2^{m^2+m}$
3. $2^{m^2-m}$
4. $2^m$

Q. If a relation $R$ is defined on set of real numbers as $xRy⟺x-y+\sqrt{2}$ is an irrational number, then the relation $R$ is

1. a reflexive relation
2. a symmetric relation
3. a transitive relation
4. both reflexive and transitive relation

Q. If $g\left(x\right)=f\left(x\right)-1$ and $f\left(x\right)+f(1-x)=2,\forall x\in R$, then $g\left(x\right)$ is symmetrical about

1. the origin
2. the line $x=\frac{1}{2}$
3. the point $(1,0)$
4. the point $(\frac{1}{2},0)$

Q. For real numbers $x$ and $y,$ a relation $R$ is defined as $xRy\iff x-y+\sqrt{2}$ is an irrational number. Then the relation $R$ is

1. reflexive and symmetric only
2. not reflexive but symmetric
3. reflexive but not symmetric and transitive
4. reflexive and transitive but not symmetric

Q. Let A = {a, b, c} and the relation R be defined on A as follows: R = {(a, a), (b, c), (a, b)}. Then, write minimum number of ordered pairs to be added in R to make it reflexive and transitive. [NCERT EXEMPLAR]

Q. Let $R$ be a reflexive relation on a finite set $A$ having $n$ elements, and let there be $m$ ordered pairs in $R,$ then:

1. $m\ge n$
2. $m\le n$
3. $m=n$
4. $m<n$

Q. Is it true that the every relation which is symmetric and transitive is also reflexive ? Give reasons.

Q. On the set $N$ of all natural numbers, define the relation $R$ by $aRb$ iff the $\text{G.C.D.}$ of $a$ and $b$ is $2$. Then the relation $R$ is

1. reflexive and symmetric
2. symmetric only
3. reflexive and transitive
4. equivalence relation