Equivalence Relation

Q.

$R$ is a relation over the set of real numbers and it is given by $nm\ge 0$. Then $r$ is

1. Symmetric and transitive

2. Reflexive and symmetric

3. A partial order relation

4. An equivalence relation

Q.

Write an example of transitive relation.

Q. Let $N$ be a set of all natural numbers and let $R$ be a relation on $N\times N$ defined by $(a,b)R(c,d)\iff ad=bc\forall (a,b),(c,d)\in N\times N.$ Then $R$ is

1. Reflexive but not transitive relation.
2. Reflexive but not symmetric relation.
3. Reflexive and symmetric relation.
4. Transitive relation

Q. Let $N$ be the set of all natural numbers and let $R$ be a relation on $N\times N$ defined by $(a,b)R(c,d)⟺ad=bc\forall (a,b),(c,d)\in N\times N.$ Then $R$ is

1. Reflexive but not transitive relation.
2. Reflexive but not symmetric relation.
3. Reflexive and symmetric relation.
4. Transitive relation

Q. Let $L$ be the set of all straight lines in the Euclidean plane. Two lines $l_1$ and $l_2$ are said to be related by the relation $R$ if $l_1$ is parallel to $l_2.$ Then the relation $R$ is

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

Q. For real numbers $x$ and $y$, we define $xRy$ iff $x-y+\sqrt{5}$ is an irrational number. The relation $R$ is

1. reflexive
2. symmetric
3. transitive
4. None of these

Q. Which among the following relations on $Z$ is an equivalence relation

1. $xRy\iff |x|=|y|$
2. $xRy\iff x\ge y$
3. $xRy\iff x>y$
4. $xRy\iff x<y$

Q.

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

1. Less than or equal to n

2. Less than n

3. Greater than or equal to n

4. None of these

Q. The relation $R$ on the set of natural numbers $N$ is defined as $xRy⟺x^2-4xy+3y^2=0;x,y\in N$ then $R$ is

1. reflexive but neither symmetric nor transitive relation.
2. symmetric but neither reflexive nor transitive relation
3. transitive but neither reflexive nor symmetric relation
4. an equivalence relation

Q. The relation $R$ on $R$ defined as $R=\left\{\right(a,b):a\le b\},$ is

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

Q. Let $A=\{2,3,4,5,\dots ,30\}$ and ${}^{\prime }{\cong }^{\prime }$ be an equivalence relation on $A\times A,$ defined by $(a,b)\cong (c,d),$ if and only if $ad=bc.$ Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :

1. $6$
2. $7$
3. $5$
4. $8$

Q. Consider the following relations:
R = {(x, y) | x, y are real numbers and x = wy for some rational number w};
S = {$(\frac{m}{n},\frac{p}{q})$| m, n, p and q are integers such that n, q $\ne$0 and qm = pn}.
Then

1. Neither R nor S is an equivalence relation
2. S is an equivalence relation but R is not an equivalence relation.
3. R and S both are equivalence relations
4. R is an equivalence relation but S is not an equivalence relation

Q. Let $L$ be the set of all straight lines in the Euclidean plane. Two lines $l_1$ and $l_2$ are said to be related by the relation $R$ if $l_1$ is parallel to $l_2.$ Then the relation $R$ is

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

Q. Let $R$ be a relation on the set of natural numbers $N$ defined by $xRy$ iff $2x+3y=21.$ Then $R$ is

1. reflexive
2. symmetric
3. transitive
4. None of these

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.

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

1. Less than n

2. Greater than or equal to n

3. Less than or equal to n

4. None of these

Q. The relation R defined on the set N of natural numbers by $xRy\iff 2x^2-3xy+y^2=0$ is

1. symemtric but not reflexive
2. only symmetric
3. not symmetric but reflexive
4. Reflexive and symmetric

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. Let R be a relation over the set $N\times n$ and it is defined by (a, b) R (c, d) $\Rightarrow$ a+ d = b + c. Then, R is

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

Q. For the given sets with $U=\{1,2,3,4,5,6,7,8,9\},$ choose the correct pair of the set with it's complement.

1. $\{1,2,5,6,7\}$
2. $\{6,5,8,7,3\}$
3. $\{1,2,5,4,8\}$
4. $\{1,2,4,9\}$
5. $\{3,6,7,9\}$
6. $\{3,4,8,9\}$

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

1. Reflexive
2. Symmetric
3. Transitive
4. None of these

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. For any two real numbers $\theta$ and $\varphi$ where $\theta ,\varphi \in (\frac{-\pi }{2},\frac{\pi }{2}),$ we define $\theta R\varphi$ if and only if ${\sec }^2\theta -{\tan }^2\varphi =1.$ Then relation $R$ is

1. Reflexive but not transitive relation.
2. Symmetric but not reflexive relation.
3. Both reflexive and symmetric relation but not transitive relation.
4. An equivalence relation

Q. For the given sets with $U=\{1,2,3,4,5,6,7,8,9\},$ choose the correct pair of the set with it's complement.

1. $\{1,2,5,6,7\}$
2. $\{6,5,8,7,3\}$
3. $\{1,2,5,4,8\}$
4. $\{1,2,4,9\}$
5. $\{3,6,7,9\}$
6. $\{3,4,8,9\}$