Relations Test 8

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Relations Test 8

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Weekly Quiz Competition

Question 1
1 / -0

Let $R$ be a relation on $N$ defined by $x+2y=8$ . The domain of $R$ is

A
$\left\{ 2,4,8 \right\}$

B
$\left\{ 2,4,6,8 \right\}$

C
$\left\{ 2,4,6 \right\}$

D
$\left\{ 1,2,3,4 \right\}$

Solution

$x+2y=8$
$x=8-2y$
Since y and x both must be natural,
Possible values of y are 1,2,3 which corresponds to possible value of x to be 6,4,2 .
Thus range of R = $\{2,4,6\}$
Question 2
1 / -0

Let $A=\left\{ 1,2,3 \right\}$ and consider the relation $R=\left\{ \left( 1,1 \right) ,\left( 2,2 \right) \left( 3,3 \right) ,\left( 1,2 \right) ,\left( 2,3 \right) ,\left( 1,3 \right) \right\}$ , then $R$ is

A
reflexive but not symmetric

B
reflexive but not transitive

C
symmetric and transitive

D
neither symmetric nor transitive

Solution

Relation $R$ is reflexive relation over set $A$ as every element of $A$ is related to itself in $R$ .

Relation $R$ is not symmetric as $(1,2)$ is in $R$ but $(2,1)$ is not in $R$ .

Relation $R$ is also transitive as for $a,b,c$ in $A$ , if $(a,b)$ is in $R$ and $(b,c)$ is in $R$ , then $(a,c)$ is also in $R$ .

Thus $A$ is correct answer .
Question 3
1 / -0

In the set $Z$ of all integers, which of the following relation $R$ is an equivalence relation?

A
$xRy:$ if $x\le y$

B
$xRy:$ if $x-= y$

C
$xRy:$ if $x- y$ is an even integer

D
$xRy:$ if $x\equiv y$ (mod 3)

Solution

For option C, xRy:if x-y is an even integer .

Let a be any number in Z,
Then, a-a=0 is an even integer .
So, Every number is related to itself .
Thus the relation is reflexive .

Let a,b be two numbers in Z,
Also let $a-b=k$ be an even integer (which implies aRb)
Then, $b-a=-k$ is also an even integer (which implies bRa).
Thus, It is evident $aRb\leftrightarrow bRA$
Thus the relation is also symmetric .

Let a,b,c be three numbers in Z,
Also let aRb and bRc,
then $a-b=2k_1$ and $b-c=2k_2$ for $k_1,k_2$ belonging to Z
which gives $a-b+b-c=2k_1+2k_2\Rightarrow a-c=2(k_1+k_2)$
which implies $aRc$ .
Thus relation is also transitive .

Thus we finally arrive at a conclusion that relation is Equivalence relation .
Question 4
1 / -0

Let $L$ denote the set of all straight lines in a plane, Let a relation $R$ be defined by $lRm$ , iff $l$ is perpendicular to $m$ for all $l \in L$ . Then, $R$ is

A
reflexive

B
symmetric

C
transitive

D
none of these

Solution

We know that if line l is perpendicular to line m, line m must also be perpendicular to line l,
So, $lRm \Leftrightarrow mRl$ and thus Relation R is symmetric .
Question 5
1 / -0

Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $aRb$ , if $a$ is congruent to $b$ for all $a,b\in T$ . Then, $R$ is

A
reflexive but not symmetric

B
transitive but not symmetric

C
equivalence

D
none of these

Solution

Let there be three triangles a,b,c in Euclidean plane such that aRb and bRc.
We know that if triangle a is congruent to triangle b and triangle b is congruent to triangle c, triangle a must also be congruent to triangle c .
So, aRc and thus relation R is transitive .

Also we know that any triangle (say t) in Euclidean plane is always congruent to itself .
So, tRt and thus relation R is reflexive .

Let there be two triangles n and m in Euclidean plane.
We know that if n is congruent to m, m must also be congruent to n.
So, $nRm \Leftrightarrow mRn$ and thus relation R is symmetric .

Thus, finally we arrive at conclusion that realation R is equivalence relation .
Question 6
1 / -0

Let $R$ be a relation on the set $N$ of natural numbers defined by $nRm$ , iff $n$ divides $m$ . Then, $R$ is

A
Reflexive and symmetric

B
Transitive and symmetric

C
Equivalence

D
Reflexive, transitive but not symmetric

Solution

Let there be a natural number $n$ ,
We know that $n$ divides $n$ , which implies $nRn$ .
So, Every natural number is related to itself in relation $R$ .
Thus relation $R$ is reflexive .

Let there be three natural numbers $a,b,c$ and let $aRb,bRc$
$aRb$ implies $a$ divides $b$ and $bRc$ implies $b$ divides $c$ , which combinedly implies that $a$ divides $c$ i.e. $aRc$ .
So, Relation R is also transitive .

Let there be two natural numbers $a,b$ and let $aRb$ ,
$aRb$ implies $a$ divides $b$ but it can't be assured that $b$ necessarily divides $a$ .
For ex, $2R4$ as 2 divides 4 but 4 does not divide 2 .
Thus Relation R is not symmetric .
Question 7
1 / -0

Let $A=\left\{ 1,2,3 \right\}$ . Then, the number of equivalence relations containing $(1,2)$ over set A is

A
$1$

B
$2$

C
$3$

D
$4$

Solution

Relation to be equivalent firstly must contain (1,1),(2,2),(3,3) to make it reflexive relation over given set .
Also since (1,2) is to be contained in set (2,1) must also be there to make relation symmetric .
So, one option of relation R is $R=\{(1,1)(2,2)(3,3)(1,2)(2,1)\}$
Now if we add (1,3) we have to also add (3,1) to keep relation symmetric .
So, second option of relation R is $R=\{(1,1)(2,2)(3,3)(1,2)(2,1)(1,3)(3,1)\}$
And, third option of relation R is $R=\{(1,1)(2,2)(3,3)(1,2)(2,1)(1,3)(3,1)(2,3)(3,2)\}$

Thus, there can be three equivalence relations containing (1,2).
Question 8
1 / -0

Let $A=\left\{ 1,2,3 \right\}$ and $R=\left\{ \left( 1,2 \right) ,\left( 2,3 \right) ,\left( 1,3 \right) \right\}$ be a relation on $A$ . Then $R$ is

A
neither reflexive nor transitive

B
neither symmetric nor transitive

C
transitive

D
none of these

Solution

The given relation is not reflexive as $(1,1),(2,2),(3,3)\not \in R$ .
Again the relation is not symmetric as $(1,2)\in R$ but $(2,1)\not \in R$ .
But the relation is transitive as $(1,2)\in R,(2,3)\in R\Rightarrow (1,2)\in R$ for all $1,2,3\in A$ .
Question 9
1 / -0

If $A=\left\{ 1,2,3 \right\}$ , then a relation $R=\left\{ \left( 2,3 \right) \right\}$ on $A$ is

A
symmetric and transitive only

B
symmetric only

C
transitive only

D
none of these

Solution

A binary relation R over Set X is transitive if for all elements a,b,c in X , if a is related to b and b is related to c, then a is related to c.
A binary relation R over Set X is symmetric iff $\forall a,b \in X (aRb \leftrightarrow bRa)$
But Given relation is not symmetric as it contains (2,3) but do not contain (3,2).
Since Given relation R contains only one pair (2,3), it is transitive .
Question 10
1 / -0

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

A
symmetric only

B
reflexive only

C
an equivalence relation

D
transitive only

Solution

The relation R is reflexive relation over set A as every element of set A is related to itself in relation R.
The relation R is also symmetric as for a,b in A, if (a,b) is in R, then (b,a) is also in R .
The relation R is also transitive as for a,b,c in A, if (a,b) is in R and (b,c) is in R, then (a,c) is also in R.

Thus relation R is an equivalence relation, So C is correct answer .

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Relations Test 8

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Relations Test 8

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Question 1

{2,4,8}\left\{ 2,4,8 \right\} {2,4,8}

{2,4,6,8}\left\{ 2,4,6,8 \right\} {2,4,6,8}

{2,4,6}\left\{ 2,4,6 \right\} {2,4,6}

{1,2,3,4}\left\{ 1,2,3,4 \right\} {1,2,3,4}

Solution

Question 2

reflexive but not symmetric

reflexive but not transitive

symmetric and transitive

neither symmetric nor transitive

Solution

Question 3

xRy:xRy:xRy: if x≤yx\le yx≤y

xRy:xRy:xRy: if x−=yx-= yx−=y

xRy:xRy:xRy: if x−yx- yx−y is an even integer

xRy:xRy:xRy: if x≡yx\equiv yx≡y (mod 3)

Solution

Question 4

reflexive

symmetric

transitive

none of these

Solution

Question 5

reflexive but not symmetric

transitive but not symmetric

equivalence

none of these

Solution

Question 6

Reflexive and symmetric

Transitive and symmetric

Equivalence

Reflexive, transitive but not symmetric

Solution

Question 7

111

222

333

444

Solution

Question 8

neither reflexive nor transitive

neither symmetric nor transitive

transitive

none of these

Solution

Question 9

symmetric and transitive only

symmetric only

transitive only

none of these

Solution

Question 10

symmetric only

reflexive only

an equivalence relation

transitive only

Solution

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$\left\{ 2,4,8 \right\}$

$\left\{ 2,4,6,8 \right\}$

$\left\{ 2,4,6 \right\}$

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

$xRy:$ if $x\le y$

$xRy:$ if $x-= y$

$xRy:$ if $x- y$ is an even integer

$xRy:$ if $x\equiv y$ (mod 3)

$1$

$2$

$3$

$4$