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

1. True
2. False

Q. 9.Let S be the set of all real numbers and let R ={(a, b) :a, b belongs to S and a=+-b}. Show that R is an equivalence relation on S.

Q.

What universal set (s) would you propose for each of the following?

(i) The set of right triangles

(ii) The set of isosceles triangles.

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. Reflexive
2. symmetric
3. transitive
4. an equivalence relation

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

1. Reflexive relation only
2. Reflexive and Symmetric relation only
3. Equivalence relation
4. Reflexive but not Transitive relation

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

Q. Let $A=\{2,3,4,5,...,17,18\}.$ 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. 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. $x{R}_{3}y\iff \frac{x}{y}$
4. $x{R}_{4}y\iff x<y$

Q. On the set $Z$ of all integer numbers, define the relation $R$ by $aRb$ iff $a+2b$ is divided by $3$. Then the relation $R$ is

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

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