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Which type of relation $ {x^2} = xy $ is:

(A) Symmetric

(B) Reflexive and transitive

(C) Transitive

(D) None of the above

(A) Symmetric

(B) Reflexive and transitive

(C) Transitive

(D) None of the above

Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = {a, b, c}. Then, R is

(a)identity relation

(b)reflexive

(c)symmetric

(d)equivalence

(a)identity relation

(b)reflexive

(c)symmetric

(d)equivalence

If R is a symmetric relation on a set A, then write a relation between R and ${{R}^{-1}}$.

Let ‘R’ be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a,): both ‘a’ and ‘b’ are either odd or even}. Show that ‘R’ is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other but no element of the subset {1, 3, 5, 6} is related to any element of the subset {2, 4, 6}.

A relation R is defined from a set $A = \left\{ {2,3,4,5} \right\}$ to a set $B = \left\{ {3,6,7,10} \right\}$ as follows: $\left( {x,y} \right) \in R \Leftrightarrow x$ is relatively prime to $y$ . Express R as a set of ordered pairs and determine its domain and range.

Write the coefficient of ${x}^{2}$ in each of the following:-

a)\[17\text{ }-\text{ }2x\text{ }+\text{ }7{{x}^{2}}\]

b)\[9\text{ }-\text{ }12x\text{ }+\text{ }{{x}^{3}}\]

c)\[\dfrac{\pi }{6}~{{x}^{2}}-\text{ }3x\text{ }+\text{ }4\]

d)\[\sqrt{3}x-\text{ }\text{ }7\]

a)\[17\text{ }-\text{ }2x\text{ }+\text{ }7{{x}^{2}}\]

b)\[9\text{ }-\text{ }12x\text{ }+\text{ }{{x}^{3}}\]

c)\[\dfrac{\pi }{6}~{{x}^{2}}-\text{ }3x\text{ }+\text{ }4\]

d)\[\sqrt{3}x-\text{ }\text{ }7\]

Let we have the sets as \[A = \left\{ {1,2,3} \right\},B = \left\{ {1,3,5} \right\}\]. If relation R from A to B is given by \[R = \left\{ {\left( {1,3} \right),\left( {2,5} \right),\left( {3,3} \right)} \right\}\]. Then \[{R^{ - 1}}\] is

1. \[\left\{ {\left( {3,3} \right),\left( {3,1} \right),\left( {5,2} \right)} \right\}\]

2. \[\left\{ {\left( {1,3} \right),\left( {2,5} \right),\left( {3,3} \right)} \right\}\]

3. \[\left\{ {\left( {1,3} \right),\left( {5,2} \right)} \right\}\]

4. None of these

1. \[\left\{ {\left( {3,3} \right),\left( {3,1} \right),\left( {5,2} \right)} \right\}\]

2. \[\left\{ {\left( {1,3} \right),\left( {2,5} \right),\left( {3,3} \right)} \right\}\]

3. \[\left\{ {\left( {1,3} \right),\left( {5,2} \right)} \right\}\]

4. None of these

If a relation R on the set \[\left\{ {1,2,3} \right\}\] be defined by \[R = \left\{ {\left( {1,2} \right)} \right\}\] , the R is

\[\left( 1 \right)\] reflexive

\[\left( 2 \right)\] transitive

\[\left( 3 \right)\] symmetric

\[\left( 4 \right)\] none of these

\[\left( 1 \right)\] reflexive

\[\left( 2 \right)\] transitive

\[\left( 3 \right)\] symmetric

\[\left( 4 \right)\] none of these

Consider the following relations:

$R = ${$(x,y)|x,y$ are real numbers and $x = wy$ for some rational number $w$}

$S = ${$\left( {\dfrac{m}{n},\dfrac{p}{q}} \right):m,n,p,q$ are integers such that $n,q \ne 0,qm = pn$}, then:

A. neither $R$ nor $S$ is an equivalence relation.

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

C. $R$ and $S$ both are the equivalence relations.

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

$R = ${$(x,y)|x,y$ are real numbers and $x = wy$ for some rational number $w$}

$S = ${$\left( {\dfrac{m}{n},\dfrac{p}{q}} \right):m,n,p,q$ are integers such that $n,q \ne 0,qm = pn$}, then:

A. neither $R$ nor $S$ is an equivalence relation.

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

C. $R$ and $S$ both are the equivalence relations.

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

If $R = \{ (x,y):x,y \in Z,{x^2} + {y^2} \leqslant 4\} $ is a relation defined on Z, Write domain of R.

Let R = {(1, 3), (2, 2), (3, 2)} and S = {(2, 1), (3, 2), (2, 3)} be two relations on set A = {1, 2, 3}. Then $SoR$ =?

(a) {(1, 3), (2, 2), (3, 2), (2, 1), (2, 3)}

(b) {(3, 2), (1, 3)}

(c) {(2, 3), (3, 2), (2, 2)}

(d) {(3, 2), (2, 1), (2, 3)}

(a) {(1, 3), (2, 2), (3, 2), (2, 1), (2, 3)}

(b) {(3, 2), (1, 3)}

(c) {(2, 3), (3, 2), (2, 2)}

(d) {(3, 2), (2, 1), (2, 3)}

If A = {1, 2, 3} and B = {1, 4, 6, 9} and R is a relation from A to B defined by ‘x’ is greater than ‘y’. The range of R is

1). {1, 4, 6, 9}

2). {4, 6, 9}

3). {1}

4). None of these

1). {1, 4, 6, 9}

2). {4, 6, 9}

3). {1}

4). None of these

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