Relations Test 13

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Relations Test 13

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Weekly Quiz Competition

Question 1
1 / -0

Let $$A=\left \{ 1, 2, 3 \right \}$$. Which of the following is not an equivalence relation on A?

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

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

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

D
$$\left \{ (1, 1),(2, 1) \right \}$$

Solution

every relation in options follow the equivalence property.
but in fourth option this option only contains two elements $$[[{1,1}],[{2,1}]]$$
this option doesn't satisfy reflexive i.e., doesn't contain all $$(a,a)$$ type $$(2,2),(3,3)$$
Not symmetric i.e., doesn't contain this type of elements $$(1,2)$$ missing
and as well transitive.

and all others options satisfy these relations.
Question 2
1 / -0

The relation "congruence modulo $$m$$" is:

A
reflexive only

B
symmetric only

C
transitive only

D
an equivalence relation

Solution

If $$R$$ be the relation, then
$$x R y$$ $$\Leftrightarrow x-y$$ is divisible by $$m$$.
$$x R x$$ because $$x-x=0$$ is divisible by $$m$$.
So, $$R$$ is reflexive.
If $$xRy\Rightarrow x-y$$ is divisible by $$m$$
$$y-x=-(x-y)$$ is divisible by $$m$$
$$\Rightarrow yRx$$
So, $$R$$ is symmetric
Let $$x R y$$ and $$y R z$$
$$\Rightarrow x-y=k_{1}m, y-z=k_{2}m$$
$$\therefore x-z=x-y+y-z=k_{1}m+k_{2}m=(k_{1}+k_{2})m$$.
So, $$R$$ is transitive.
As R is reflexive, symmetric and transitive, it is an equivalence relation.
Question 3
1 / -0

The minimum number of elements that must be added to the relation $$R=\left \{ (1, 2),(2, 3) \right \}$$ on the set of natural numbers, so that it is an equivalence is:

A
$$4$$

B
$$7$$

C
$$6$$

D
$$5$$

Solution

Since $$R=\left \{ (1, 2),(2, 3) \right \}$$
i.e., $$A=\left \{ (1, 2) \right \} $$ and $$B=\left \{ 2, 3 \right \}$$
Now, if $$R_{r}$$ is the reflexive relation, such that
$$R_{1}=\left \{ (1, 2),(2, 3),(1, 1),(2, 2),(3, 3) \right \}$$ has $$5$$ elements.
Now, if R' is both symmetric and reflexive relation, then
$$R_{2}=\left \{ (1, 2),(2, 3),(1, 1),(2, 2),(2, 1),(3, 2),(3, 3) \right \}$$ has $$7$$ elements
Again, if $$R_{3}$$ is reflexive, symmetric and transitive all together, than
$$R_{3}=\begin{Bmatrix} (1, 2) & (2, 3) & (1, 1) \\ (2, 2) & (2, 1) & (3, 2) \\ (3, 3) & (1, 3) & (3, 1)\end{Bmatrix} $$
has $$9$$ elements. Starting from $$2$$ elements, therefore, the minimum number of elements to be added is $$7$$.
Question 4
1 / -0

Let R be the relation in the set N given by $$=\left \{ (a, b):a=b-2, b >6 \right \}$$. Choose the correct answer.

A
$$(2, 4)\in R$$

B
$$(3, 8)\in R$$

C
$$(6, 8)\in R$$

D
$$(8, 7)\in R$$

Solution

$$(a,b)∈R$$ only if $$a=b−2$$ and $$b>6$$
$$(6,8)∈R$$ as $$6=8−2$$ and $$8 >6$$
Hence $$(c)$$ is the correct alternative
Question 5
1 / -0

Directions For Questions

Based on this information answer the question
(i) If $$A=\left \{ 1, 2, 3, 4 \right \}$$ the $$p(A)=\left \{ \phi , \left \{ 1 \right \},
\left \{ 2 \right \}, \left \{ 3 \right \}, \left \{ 1, 2 \right \}, \left \{ 2, 3 \right \},
\left \{ 1, 3 \right \}, \left \{ 1, 2, 3 \right \} \right \}$$
Power set of an empty set, $$p(\phi )=\left \{ \left ( \phi \right ) \right \}$$ and $$n\{p(\phi
)\}=1.$$
(ii) Any subset of $$A\times A $$ is called a relation on A. Number of relations on $$A=2^{n^{2}}$$

...view full instructions

Find the number of relations from $$\left \{ m, o, t, h, e, r \right \}$$ to $$\left \{ c, h, i, l, d \right \}$$

A
$$30$$

B
$$30^{2}$$

C
$$2^{30}$$

D
$$60$$

Solution

$$n \{m.o,t,b,e,r\} = 6 = m$$ (let)
$$n \{c,h,i,l,d\} = 5 = n$$ (let)

$$\therefore $$ numbers of Relations
$$= { 2 }^{ mn }$$
$$= { 2 }^{ 6\times 5 }$$
$$= { 2 }^{ 30 }$$
Question 6
1 / -0

If $$R$$ is a relation from a set $$A$$ to a set $$B$$ and $$S$$ is a relation from $$B$$ to a set $$C$$, then the relation $$SOR$$

A
is from $$A$$ to $$C$$

B
is from $$C$$ to $$A$$

C
does not exist

D
none of these

Solution

Domain of $$SOR$$ is the domain of $$R$$ and the range is the range of $$S$$. Hence the mapping is from set $$A$$ to set $$C$$.
Question 7
1 / -0

Let $$A=\left \{ 1,2,3 \right \}$$. The total number of distinct relations that can be defined over $$A$$ is:

A
$$2^{9}$$

B
$$6$$

C
$$8$$

D
None of the above

Solution

Total number of distinct binary relations over the set $$ A$$ will be
$$2^{n^2}$$
$$=2^{3^{2}}$$
$$=2^{9}$$
$$=512$$
Question 8
1 / -0

Which of the following statements is true?

A
The x-axis is a vertical line

B
The y-axis is a horizontal line

C
The scale on both axes must be the same in a Cartesian plane

D
The point of intersection between the x-axis and the y-axis is called the origin

Solution

Since, Origin is the point of intersection of $$x$$ and $$y,$$ we can say that option $$D$$ is correct.
Question 9
1 / -0

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 $$R$$o$$S$$ is equal

A
$$\{(2, 3), (3, 2), (2, 2)\}$$

B
$$\{(1, 3), (2, 2), (3, 2), (2 ,1), (2, 3)\}$$

C
$$\{(3, 2), (1, 3)\}$$

D
$$\{(2, 3), (3, 2)\}$$

Solution

If $$f\epsilon X\times Y$$ and $$g\epsilon Y\times Z$$

Then
$$gof\epsilon X\times Z$$ i.e
$$ (x,z)\epsilon gof$$

Applying the above concept, we gives us
$$RoS=\{(2,1)|(1,3),(3,2)|(2,2),(2,3)|(3,2)\}$$

$$=\{(2,3),(3,2),(2,2)\}$$
Question 10
1 / -0

If $$A=\{1, 2, 3\}$$ and $$B=\{3, 8\}$$, then $$(A\cup B)\times (A\cap B)$$ is

A
$$\{(3, 1), (3, 2), (3, 3), (3, 8)\}$$

B
$$\{(1, 3), (2, 3), (3, 3), (8, 3)\}$$

C
$$\{(1, 2), (2, 2), (3, 3), (8, 8)\}$$

D
$$\{(8, 3), (8, 2), (8, 1), (8, 8)\}$$

Solution

$$A\cup B=\{1,2,3,8\}$$
$$A\cap B=\{3\}$$
$$\therefore (A\cup B)\times (A\cap B)$$
$$=\{(1,3),(2,3),(3,3),(8,3)\}$$

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Re-Attempt Weekly Quiz Competition

Relations Test 13

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Relations Test 13

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SHARING IS CARING

Question 1

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

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

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

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

Solution

Question 2

reflexive only

symmetric only

transitive only

an equivalence relation

Solution

Question 3

$$4$$

$$7$$

$$6$$

$$5$$

Solution

Question 4

$$(2, 4)\in R$$

$$(3, 8)\in R$$

$$(6, 8)\in R$$

$$(8, 7)\in R$$

Solution

Question 5

$$30$$

$$30^{2}$$

$$2^{30}$$

$$60$$

Solution

Question 6

is from $$A$$ to $$C$$

is from $$C$$ to $$A$$

does not exist

none of these

Solution

Question 7

$$2^{9}$$

$$6$$

$$8$$

None of the above

Solution

Question 8

The x-axis is a vertical line

The y-axis is a horizontal line

The scale on both axes must be the same in a Cartesian plane

The point of intersection between the x-axis and the y-axis is called the origin

Solution

Question 9

$$\{(2, 3), (3, 2), (2, 2)\}$$

$$\{(1, 3), (2, 2), (3, 2), (2 ,1), (2, 3)\}$$

$$\{(3, 2), (1, 3)\}$$

$$\{(2, 3), (3, 2)\}$$

Solution

Question 10

$$\{(3, 1), (3, 2), (3, 3), (3, 8)\}$$

$$\{(1, 3), (2, 3), (3, 3), (8, 3)\}$$

$$\{(1, 2), (2, 2), (3, 3), (8, 8)\}$$

$$\{(8, 3), (8, 2), (8, 1), (8, 8)\}$$

Solution

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