Relations Test 9

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Relations Test 9

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Weekly Quiz Competition

Question 1
1 / -0

Let $$R$$ be a reflexive relation on a finite set $$A$$ having $$n$$ elements, and let there be $$m$$ ordered pairs in $$R,$$ then:

A
$$m\geq n$$

B
$$m\leq n$$

C
$$m=n$$

D
$$m< n$$

Solution

By the definition for a relation with $$n$$ elements to be reflexive relation it must have at least $$n$$ ordered pairs.
it has $$m$$ ordered pairs therefore $$\mathrm{m}\geq \mathrm{n}$$.
Question 2
1 / -0

Total number of equivalence relations defined in the set $$S =\{a,b,c\}$$ is

A
$$5$$

B
$$3$$!

C
$$2^{3}$$

D
$$3^{3}$$

Solution

The smallest equivalence relation is the identity relation $$R_1=\{(a,a),(b,b),(c,c)\}$$
Then two ordered pairs of two distinct elements can be added to give three more equivalence relations.
$$R_2=\{(a,a),(b,b),( c,c),(a,b),(b,a)\}$$
Similarly $$R_3$$and $$R_4$$ can be made by taking $$(b,c),(c,b)$$ and $$(a,c),(c,a)$$ respectively.
Finally the largest equivalence relation i.e., the universal relation
$$R_5=\{(a,a),(b, b),(c, c),(a, b),(b, a),(a, c),(c,a ),(b, c),(c, b) \}$$
Hence, total $$5$$ equivalence relations can be created.
Question 3
1 / -0

If relation $$R$$ is defined by $$\mathrm{R}=\{(\mathrm{x},\ \mathrm{y}):2\mathrm{x}^{2}+3\mathrm{y}^{2}\leq 6\}$$, then the domain of $$\mathrm{R}$$ is

A
$$[-3,3]$$

B
$$[-\sqrt{3},\sqrt{3}]$$

C
$$[-\sqrt{2},\sqrt{2}]$$

D
$$[-2,2]$$

Solution

Domain $$=$$ value of $$\mathrm{x}$$ in $$f=2\mathrm{x}^{2}+3\mathrm{y}^{2}\leq 6$$ then
$$2\mathrm{x}^{2}+3\mathrm{y}^{2}\leq 6$$ = $$\displaystyle \frac{\mathrm{x}^{2}}{3}+\frac{\mathrm{y}^{2}}{2}\leq 1$$
Since it is an equation of ellipse then value of $$x$$ $$ \Rightarrow \mathrm{x}\in[-\sqrt{3},\sqrt{3}]$$
Question 4
1 / -0

The domain and range of relation $$R=\{(x,y) | x, y \in N$$, $$x+2y=5\} $$ is?

A
$$\{1,3\}, \{2,1\}$$

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

C
$$\{1,3\}, \{1,1\}$$

D
$$\{1,2\}, \{1,3\}$$

Solution

The possible values of $$x$$ which satisfies the given relation is the domain of the relation
The possible values of $$y$$ which satisfies the given relation is the range of the relation
Given that $$x,y$$ are natural numbers and the relation is $$x+2y=5$$
For $$x=1$$ , the value of $$y$$ is $$2$$
For $$x=3$$ , the value of $$y$$ is $$1$$
If $$x\geq 5$$, then $$y$$ is negative which does not belong to naturals.
If $$x=2,4$$ then $$y$$ becomes rational number which is not in naturals.
Thus, only $$1$$ and $$3$$ gives $$y$$ as natural numbers $$2$$ and $$1$$.
Therefore, the domain is $$\{1,3\}$$ and the range is $$\{2,1\}$$
Question 5
1 / -0

Let $$A$$ be a non-empty set such that $$A \times A$$ has $$9 $$ elements among which are found $$(-1, 0)$$ and $$(0, 1)$$, then

A
$$A=\left\{ -1,0 \right\}$$

B
$$A=\left\{ 0,1 \right\}$$

C
$$A=\left\{ -1,0,1 \right\}$$

D
$$A=\left\{ -1,1 \right\}$$

Solution

Given $$A\times A$$ has $$9$$ elements $$\Rightarrow n(A)=3$$
Elements of $$A\times A $$ are $$(-1,0)$$ & $$(0,1)$$.
$$\therefore$$ $$A$$ needs to contain $$-1, 0$$ and $$1$$.
$$\therefore$$ Since $$n(A)=3 \Rightarrow A=\left \{ -1,0,1 \right \}$$
Question 6
1 / -0

Given the relation $$R =\{(1,2) (2,3)\}$$ on the set $$A=\{1,2,3\}$$, the minimum number of ordered pairs which when added to $$R$$ to make it an equivalence relation is

A
$$5$$

B
$$6$$

C
$$7$$

D
$$8$$

Solution

$$R$$ is reflexive if it contains $$(1, 1), (2, 2), (3, 3)$$.
$$\because (1, 2)\in R$$ and $$ (2,3)\in R$$
$$\therefore R$$ is symmetric if $$(2, 1), (3, 2) \in R.$$
As $$(1, 2), (2,3)\in R$$
Then, $$R$$ is transitive is $$(1,3)\in R$$
Also, $$(3,1)\in R$$ for symmetric property.
Now, $$R=\{(1,1),(2,2),(3,3),(2,1),(3,2),(2,3),(1,2),(1,3),(3,1)\}$$
Thus, $$R$$ becomes an equivalence relation by adding
$$(1,1),(2,2),(3,3),(2,1),(3,2),(1,3),(3,1)$$
Hence, the total number of ordered pairs is $$7$$.
Question 7
1 / -0

Which one of the following relations on Z is equivalence relation?

A
$$xR_1 y\Leftrightarrow |x| = |y|$$

B
$$xR_2 y\Leftrightarrow x \geq y$$

C
$$xR_3 y\Leftrightarrow \dfrac{x}{y}$$

D
$$xR_4 y\Leftrightarrow x < y$$

Solution

A relation is equivalence if it is symmetric, transitive and reflexive
Consider, $$xR_1y\Leftrightarrow \left | x \right |=\left | y \right |$$
For reflexive:
$$xR_1x\Leftrightarrow \left | x \right |=\left | x \right |$$, clearly it satisfies $$R_{1}$$
Thus, $$R_{1}$$ is reflexive.
For symmetric:
$$xR_{1}y\Leftrightarrow \left | x \right |=\left | y \right |$$ $$\implies yR_{1}x\Rightarrow \left | y \right |=\left | x \right |$$ as equality communicative
So, $$ R_{1}$$ is symmetric.
For transitive:
Let $$(x,y)$$ and $$(y,z)$$ satisfy $$R_{1}$$
$$\implies xR_{1}y\Leftrightarrow \left | x \right |=\left | y \right |$$ and $$yR_{1}z\Leftrightarrow \left | y \right |=\left | z \right |$$
$$\implies |x|=|y|$$ and $$|y |=|z|$$
$$\implies |x |=|z|$$
$$\therefore xR_{1}z\Leftrightarrow \left | x \right |=\left | z \right |$$
$$\therefore \left ( x,z \right )$$ satisfies $$R_{1}$$
Thus, $$R_1$$ is a transitive relation.
Hence, $$ R_{1}$$ is equivalent.
Also, relations $$R_{2},\ R_{3}$$ and $$R_{4}$$ are not symmeteric on $$Z,$$ hence not an equivalence relations.
Question 8
1 / -0

Which of the following are not equivalence relations on $$I$$?

A
$$aRb$$ if $$a + b$$ is an even integer

B
$$aRb$$ if $$a - b$$ is an even integer

C
$$aRb$$ if $$a < b$$

D
$$aRb$$ if $$a = b$$

Solution

Option A, $$aRb$$ if $$a+b$$ is an even integer.

Let $$a, b,c \in I$$
Clearly, $$a+a =2a$$ is an even integer.
Hence, $$aRa$$ for all $$a\in I$$

Now, let $$aRb$$
$$\Rightarrow a+b$$ is an even integer.
or $$b+a$$ is an even integer.
$$\Rightarrow bRa$$

Next, let $$aRb, bRc$$
$$\Rightarrow a+b $$ in an even integer and $$ b+c$$ is an even integer
$$\Rightarrow a$$ and $$b$$ both are even or both are odd . Also, $$b$$ and $$c$$ both are even or both are odd.
Hence, if $$a$$ and $$b$$ are even .So $$c$$ is also even . Hence, $$a+c$$ is even.
And if $$a$$ and $$b$$ are odd.So, $$c$$ is also odd . Again $$a+c$$ is even.
Hence, $$aRc$$

Hence, option A is equivalence relation.

Option B, $$aRb$$ if $$a-b$$ is an even integer.

Let $$a, b,c \in I$$
Clearly, $$a-a =0$$ is an even integer.
Hence, $$aRa$$ for all $$a\in I$$

Now, let $$aRb$$
$$\Rightarrow a-b$$ is an even integer.
or $$-(a-b)$$ is also an even integer.
$$\Rightarrow \ bRa$$

Next, let $$aRb, bRc$$
$$\Rightarrow a-b $$ in an even integer and $$ b-c$$ is an even integer
$$a+c=a-b+b-c=$$ even+even
Hence, $$a+c$$ is even.
Hence, $$aRc$$

Hence, option $$B$$ is equivalence relation.

Option C $$aRb$$ if $$a<b$$

Let $$a, b,c \in I$$
Clearly,symmetry does not hold here
Now, let $$aRb$$
$$\Rightarrow a<b$$
But it does not implies $$b<a$$

Hence, option C is not an equivalence relation.

Option D, $$aRb$$ if $$a=b$$

Let $$a, b,c \in I$$
Clearly, $$a=a$$
Hence, $$aRa$$ for all $$a\in I$$

Now, let $$aRb$$
$$\Rightarrow a=b$$
or $$b=a$$ is an even integer.
$$\Rightarrow bRa$$

Next, let $$aRb, bRc$$
$$\Rightarrow a=b $$and $$ b=c$$
$$\Rightarrow a=b=c$$
Hence, $$aRc$$

Hence, option D is equivalence relation.
Question 9
1 / -0

The number of relations from $$ A=\left \{ 1,2,3 \right \} $$ to $$B = \left \{ 4,6,8,10 \right \}$$ is

A
$$4^{3}$$

B
$$2^{7}$$

C
$$ 2^{12}$$

D
$$3^{4}$$

Solution

$$n\left ( B \right )=4$$
$$n\left ( A \right )=3$$
Therefore there exists $$2^{3 \times 4}$$ relations from $$A$$ to $$B$$
i.e $$2^{12}$$ relations
Question 10
1 / -0

The range of the function $$f(x)=\dfrac{\sin(\pi [x])}{x^{2}+1}$$ (where $$[.]$$ denotes greatest integer function) is

A
$$\{0\}$$

B
$$R$$

C
$$(0,1)$$

D
$$(0,\infty )$$

Solution

$$f(x)=\dfrac {\sin(\pi[x])}{x^2+1}$$
$$\because$$ $$[x]$$ always gives integer value and $$\sin n\pi =0$$
$$\therefore f(x)=0$$

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Re-Attempt Weekly Quiz Competition

Relations Test 9

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Relations Test 9

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SHARING IS CARING

Question 1

$$m\geq n$$

$$m\leq n$$

$$m=n$$

$$m< n$$

Solution

Question 2

$$5$$

$$3$$!

$$2^{3}$$

$$3^{3}$$

Solution

Question 3

$$[-3,3]$$

$$[-\sqrt{3},\sqrt{3}]$$

$$[-\sqrt{2},\sqrt{2}]$$

$$[-2,2]$$

Solution

Question 4

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

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

$$\{1,3\}, \{1,1\}$$

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

Solution

Question 5

$$A=\left\{ -1,0 \right\}$$

$$A=\left\{ 0,1 \right\}$$

$$A=\left\{ -1,0,1 \right\}$$

$$A=\left\{ -1,1 \right\}$$

Solution

Question 6

$$5$$

$$6$$

$$7$$

$$8$$

Solution

Question 7

$$xR_1 y\Leftrightarrow |x| = |y|$$

$$xR_2 y\Leftrightarrow x \geq y$$

$$xR_3 y\Leftrightarrow \dfrac{x}{y}$$

$$xR_4 y\Leftrightarrow x < y$$

Solution

Question 8

$$aRb$$ if $$a + b$$ is an even integer

$$aRb$$ if $$a - b$$ is an even integer

$$aRb$$ if $$a < b$$

$$aRb$$ if $$a = b$$

Solution

Question 9

$$4^{3}$$

$$2^{7}$$

$$ 2^{12}$$

$$3^{4}$$

Solution

Question 10

$$\{0\}$$

$$R$$

$$(0,1)$$

$$(0,\infty )$$

Solution

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