Relations Test 34

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Relations Test 34

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Weekly Quiz Competition

Question 1
1 / -0

The relation $$R$$ defined on the set $$A=\left\{ 1,2,3,4,5 \right\} $$ by $$R=\left\{ \left( a,b \right) :\left| { a }^{ 2 }-{ b }^{ 2 } \right| <16 \right\} $$, is not given by

A
$$\left\{ \left( 1,1 \right) ,\left( 2,1 \right) ,\left( 3,1 \right) ,\left( 4,1 \right) ,\left( 2,3 \right) \right\} $$

B
$$\left\{ \left( 2,2 \right) ,\left( 3,2 \right) ,\left( 4,2 \right) ,\left( 2,4 \right) \right\} $$

C
$$\left\{ \left( 3,3 \right) ,\left( 4,3 \right) ,\left( 5,4 \right) ,\left( 3,4 \right) \right\} $$

D
none of these

Solution

Given, the relation $$R$$ defined on the set $$A=\left\{ 1,2,3,4,5 \right\} $$ by $$R=\left\{ \left( a,b \right) :\left| { a }^{ 2 }-{ b }^{ 2 } \right| <16 \right\} $$.

Then $$R$$ can be written as,
$$R=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),$$ $$(4,3),(4,4),(4,5),(5,4),(5,5)\}$$.
Question 2
1 / -0

If $$R$$ is a relation on the set $$A=\left\{ 1,2,3,4,5,6,7,8,9 \right\} $$ given by $$xRy\Leftrightarrow y=3x$$, then $$R=$$

A
$$\left\{ \left( 3,1 \right) ,\left( 6,2 \right) ,\left( 8,2 \right) ,\left( 9,3 \right) \right\} $$

B
$$\left\{ \left( 3,1 \right) ,\left( 6,2 \right) ,\left( 9,3 \right) \right\} $$

C
$$\left\{ \left( 3,1 \right) ,\left( 2,6 \right) ,\left( 3,9 \right) \right\} $$

D
none of these

Solution

Given, $$R$$ is a relation on the set $$A=\left\{ 1,2,3,4,5,6,7,8,9 \right\} $$ given by $$xRy\Leftrightarrow y=3x$$, then $$R=\{(1,3),(6,2),(9,3)\}$$.
Question 3
1 / -0

Let $$A=\left\{ 2,3,4,5,....,17,18 \right\} $$. Let $$\simeq $$ be the equivalence relation on $$A\times A$$, cartesian product of $$A$$ with itself, defined by $$(a,b)\simeq (c,d)$$, iff $$ad=bc$$. The the number of ordered pairs of the equivalence class of $$(3,2)$$ is

A
$$4$$

B
$$5$$

C
$$6$$

D
$$7$$

Solution

Let $$\left(3,2\right)\cong\left(x,y\right)$$
$$\Rightarrow\,3y=2x$$
This is possible in the cases:
$$x=3,\,y=2$$
$$x=6,\,y=4$$
$$x=9,\,y=6$$
$$x=12,\,y=3$$
$$x=15,\,y=10$$
$$x=18,\,y=12$$
Hence total pairs are $$6$$.
Question 4
1 / -0

If $$R$$ is the largest equivalence relation on a set $$A$$ and $$S$$ is any relation on $$A$$, then

A
$$R\subset S$$

B
$$S\subset R$$

C
$$R=S$$

D
none of these

Solution

$$S$$ is a subset of $$R$$
Since $$R$$ is the largest equivalence relation on set $$A$$,
$$R$$ is a subset of $$A\times A$$
Since $$S$$ is any relation $$A$$
$$S$$ is contained in $$R$$
and $$R$$ is a subset of $$a\times A$$
So, $$S\subset R$$
Question 5
1 / -0

The maximum number of equivalence relations on the set $$A=\left\{ 1,2,3 \right\} $$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$5$$

Solution

$${R}_{1}=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right)\right\}$$
$${R}_{2}=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,1\right)\right\}$$
$${R}_{3}=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,3\right),\left(3,1\right)\right\}$$
$${R}_{4}=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(2,3\right),\left(3,2\right)\right\}$$
$${R}_{5}=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,1\right),\left(1,3\right),\left(3,1\right),\left(2,3\right),\left(3,2\right)\right\}$$
These are the $$5$$ relations on $$A$$ which are equivalence.
Question 6
1 / -0

For real number $$x$$ and $$y$$, define $$xRy$$ iff $$x-y+\sqrt{2}$$ is an irrational number. Then the relation $$R$$ is

A
reflexive

B
symmetric

C
transitive

D
none of these

Solution

For each value of x ∈ R, $$x − x + \sqrt 2$$ that is $$\sqrt 2$$ is an irrational number.

It is reflexive.

Let $$x = \sqrt 2$$ and $$y =2$$ then $$x − y + \sqrt 2 = 2 \sqrt 2 – 2$$ which is irrational but when $$y = \sqrt 2$$ and $$x = 2$$, $$x − y + \sqrt 2$$ is not irrational.

It is not symmetric.

Let $$x − y + \sqrt 2$$ is irrational & $$y − z + \sqrt 2$$ is irrational then in above case let $$x = 1; y = \sqrt 2 × 2$$ & $$z = \sqrt 2$$

Hence $$x − z + \sqrt 2$$ is not irrational, so, the relation is not transitive.
Question 7
1 / -0

If $$A=\left\{ a,b,c,d \right\} $$, then a relation $$R=\left\{ \left( a,b \right) ,\left( b,a \right) ,\left( a,a \right) \right\} $$ on $$A$$ is

A
symmetric and transitive only

B
reflexive and transitive only

C
symmetric only

D
transitive only

Solution

Given if $$A=\left\{ a,b,c,d \right\} $$, then a relation $$R=\left\{ \left( a,b \right) ,\left( b,a \right) ,\left( a,a \right) \right\} $$.
Since $$(b,b)\not \in R, (c,c)\not \in R$$ so this relation is not reflexive.
The relation is symmetric as if $$(a,b)\in R\Rightarrow (b,a)\in R$$ for all $$a,b\in A$$.
The relation is not transitive also as $$(b,a)\in R,(a,b)\in R$$ but $$(b,b)\not \in R$$.
So the relation is only symmetric.
Question 8
1 / -0

The number of ordered pairs (a, b) of positive integers such that $$\dfrac{2a - 1}{b}$$ and $$\dfrac{2b - 1}{a}$$ are both integers is

A
$$1$$

B
$$2$$

C
$$3$$

D
more than $$3$$

Solution

The number of ordered pairs $$\left(a, b\right)$$ of positive integers where $$a,b\in{I}^{+}$$
$$\Rightarrow\left(\dfrac{2a-1}{b},\dfrac{2b-1}{a}\right)\in{I}^{+}$$
Let $$\dfrac{2a-1}{b}=1$$
$$\Rightarrow\,2a-1=b$$

and
$$\Rightarrow\,\dfrac{2b-1}{a}=\dfrac{2\left(2a-1\right)-1}{a}$$
$$=\dfrac{4a-2-1}{a}$$
$$=\dfrac{4a-3}{a}$$

$$\therefore \dfrac{2b-1}{a}=4-\dfrac{3}{a}$$
For integer,$$a=1,3$$ we get
$$b=2a-1=2-1=1$$ for $$a=1$$
$$b=2a-1=2\times 3-1=6-1=5$$
$$\therefore\,b=1,5$$
So total sets are $$\left(1,1\right),\left(3,5\right),\left(5,3\right)$$
Hence there are totally $$3$$ sets.
Question 9
1 / -0

Let Z be the set of all integers and let R be a relation on Z defined by $$a$$ R $$b\Leftrightarrow (a-b)$$ is divisible by $$3$$. Then, R is?

A
Reflexive and symmetric but not transitive

B
Reflexive and transitive but not symmetric

C
Symmetric and transitive but not reflexive

D
An equivalence relation

Solution

Let R={(a,b):a,b∈Z and (a−b) is divisible by 3}.

Show that R is an equivalence relation on Z.

Given:

R={(a,b):a,b∈Z and (a−b) is divisible by 3}.

$$R = (a,b)$$

$$(a-b)$$ is divisible by $$3$$

Reflexive

$$(a,a) \Rightarrow (a-a)$$ is divisible by $$3$$

Symmetric

$$(a, b)\Leftrightarrow (b, a)$$

$$(a-b)$$ is divisible by $$3$$

$$(b-a)$$ is divisible by $$3$$

Transitive

$$(a,b), (b,c)\Leftrightarrow (a,c)$$

$$(a-b)$$ is divisible by $$3$$

$$(b-c)$$ is divisible by $$3$$

$$(a-c)$$ is divisible by $$3$$

R is an equivalent relation on Z
Question 10
1 / -0

If $$A=\{2, 4, 5\}, B=\{7, 8, 9\}$$ then $$n(A\times B)$$ is equal to

A
$$6$$

B
$$9$$

C
$$3$$

D
$$0$$

Solution

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Relations Test 34

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Relations Test 34

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Question 1

$$\left\{ \left( 1,1 \right) ,\left( 2,1 \right) ,\left( 3,1 \right) ,\left( 4,1 \right) ,\left( 2,3 \right) \right\} $$

$$\left\{ \left( 2,2 \right) ,\left( 3,2 \right) ,\left( 4,2 \right) ,\left( 2,4 \right) \right\} $$

$$\left\{ \left( 3,3 \right) ,\left( 4,3 \right) ,\left( 5,4 \right) ,\left( 3,4 \right) \right\} $$

none of these

Solution

Question 2

$$\left\{ \left( 3,1 \right) ,\left( 6,2 \right) ,\left( 8,2 \right) ,\left( 9,3 \right) \right\} $$

$$\left\{ \left( 3,1 \right) ,\left( 6,2 \right) ,\left( 9,3 \right) \right\} $$

$$\left\{ \left( 3,1 \right) ,\left( 2,6 \right) ,\left( 3,9 \right) \right\} $$

none of these

Solution

Question 3

$$4$$

$$5$$

$$6$$

$$7$$

Solution

Question 4

$$R\subset S$$

$$S\subset R$$

$$R=S$$

none of these

Solution

Question 5

$$1$$

$$2$$

$$3$$

$$5$$

Solution

Question 6

reflexive

symmetric

transitive

none of these

Solution

Question 7

symmetric and transitive only

reflexive and transitive only

symmetric only

transitive only

Solution

Question 8

$$1$$

$$2$$

$$3$$

more than $$3$$

Solution

Question 9

Reflexive and symmetric but not transitive

Reflexive and transitive but not symmetric

Symmetric and transitive but not reflexive

An equivalence relation

Solution

Question 10

$$6$$

$$9$$

$$3$$

$$0$$

Solution

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