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

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

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

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

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

Solution

Question 2

reflexive but not symmetric

reflexive but not transitive

symmetric and transitive

neither symmetric nor transitive

Solution

Question 3

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

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

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

$$xRy:$$ if $$x\equiv y$$ (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

$$1$$

$$2$$

$$3$$

$$4$$

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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