Equivalence Relation

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 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. 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. 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.

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

1. Flase

2. True

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

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 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. 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. 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