Equivalence Relation

Q.

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

1. Reflexive

2. Symmetric

3. Transitive

4. Equivalence

5. All the above

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

1. relexive, symmetric and not transitive
2. relexive, symmetric and transitive
3. reflexive, not symmetric and transitive
4. 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 ____

1. 9
2. 8
3. 64
4. 6

Q.

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

1. Reflexive, symmetric but not transitive

2. Reflexive, neither symmetric nor transitive

3. Reflexive, symmetric and transitive

4. None of the above

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

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

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

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

4. $R$ and $S$ both are equivalence relations

Q. Let $R$ be a relation defined in the set of real numbers by $aRb\iff 1+ab>0$. Then $R$ is

1. Equivalence relation
2. Transitive
3. Symmetric
4. Anti-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. an equivalence relation
4. transitive only

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 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

1. symmetric
2. Reflexive
3. transitive
4. 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

1. True

2. False

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

1. reflexive
2. transitive
3. symmetric
4. 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

1. Less than n

2. Greater than or equal to n

3. Less than or equal to n

4. 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 $\iff$ |a-b| $\le$ 1. Then, R is

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

1. equivalence relation
2. partial order relation
3. reflexive but not symmetric
4. 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 = {$(\frac{m}{n},\frac{p}{q})$| m, n, p and q are integers such that n, q $\ne$0 and qm = pn}.
Then

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

Q. If we define a relation $R$ on the set $N\times N$ as $(a,b)R(c,d)\iff a+d=b+c$ for all $(a,b),(c,d)ϵN\times N$, then the relation is

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

1. Reflexive and Transitive

2. Symmetric and Transitive

3. Symmetric only

4. 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?

1. $x{R}_{1}y\iff |x|=|y|$
2. $x{R}_{3}y\iff \frac{x}{y}$
3. $x{R}_{2}y\iff x\ge y$
4. $x{R}_{4}y\iff x<y$

Q. Let R be a relation defined by R = $(a,b):a\ge b,a,bϵR$. The relation R is

1. Reflexive, symmetric and transitive
2. Reflexive, transitive but not symmetric
3. Neither transitive nor reflexive but symmetric
4. 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-

1. A set
2. An equivalence relation
3. A relation
4. None of the above

Q.

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

1. Flase

2. True

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

1. $x{R}_{1}y\iff |x|=|y|$
2. $x{R}_{2}y\iff x\ge y$
3. $x{R}_{4}y\iff x<y$
4. $x{R}_{3}y\iff \frac{x}{y}$

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

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