Relations Test 5

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Relations Test 5

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Weekly Quiz Competition

Question 1
1 / -0

Let R be the set of real numbers and the mapping $$f:R\rightarrow R$$ and $$g:R\rightarrow R$$ be defined by $$f(x)=5-x^2$$ and $$g(x)=3\lambda-4$$, then the value of $$(fog)(-1)$$ is

A
$$-44$$

B
$$-54$$

C
$$-32$$

D
$$-64$$

Solution

Given, $$f(x)=5-x^2,\,g(x)=3x-4$$
Now, $$(fog)(-1)=f\begin{Bmatrix}g(-1)\end{Bmatrix}$$
$$=f(-7)\;[\because\;g(-1)=3(-1)-4=-7]$$
$$=5-(-7)^2$$
$$=-44$$
Question 2
1 / -0

Let the number of elements of the sets $$A$$ and $$B$$ be $$p$$ and $$q$$ respectively. Then, the number of relations from the set $$A$$ to the set $$B$$ is

A
$${ 2 }^{ p+q }$$

B
$${ 2 }^{ pq }$$

C
$$p+q$$

D
$$pq$$

Solution

Given, number of elements of $$A$$ and $$B$$ are $$p$$ and $$q$$ respectively.
$$\therefore$$ The number of relations from the set $$A$$ to the set $$B$$ is $${2}^{pq}$$.
Question 3
1 / -0

Let $$A = \left \{x, y, z\right \}$$ and $$B = \left \{p, q, r, s\right \}$$. What is the number of distinct relations from $$B$$ to $$A$$?

A
$$4096$$

B
$$4094$$

C
$$128$$

D
$$126$$

Solution

Since, No. of elements in set A$$=3$$
No. of elements in set B$$=4$$
$$\therefore$$ No. of elements in B$$\times$$A$$=3\times 4=12$$
$$\therefore$$ No. of distinct relations from B to A $$= 2^{12}=4096$$
Option A is correct.
Question 4
1 / -0

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

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

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

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

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

Solution

Firstly b should be greater than 6 & also difference between b& a should be 2 , option C is satisfing both the conditions hence the correct answer is C
Question 5
1 / -0

Let A be a finite set containing n distinct elements. The number of relations that can be defined on A is.

A
$$2^n$$

B
$$n^2$$

C
$$2n^2$$

D
$$2n$$
Question 6
1 / -0

On the set $$N$$ of all natural numbers define the relation $$R$$ by $$a R b$$ if and only if the G.C.D. of $$a$$ and $$b$$ is $$2$$. Then $$R$$ is:

A
Reflexive, but not symmetric

B
Symmetric only

C
Reflexive and transitive

D
Reflexive, symmetric and transitive

Solution

$$aRa$$ only if GCD of a and a is 2 which is clearly not the case.
Hence not reflexive.

$$aRb \Rightarrow bRa$$ if GCD of a and b is 2 then GCD of b and a is also 2.
The relation is symmetric.

Transitiveness is not necessary,i.e, if and b have GCD 2 and b and c also have GCD 2 a and c need not have GCD 2.

The relation is symmetric only.
Question 7
1 / -0

Consider two sets $$A=\{a, b, c\}, B=\{e, f\}$$. If maximum numbers of total relations from A to B; symmetric relation from A to A and from B to B are $$l, m, n$$ respectively, then the value of $$2l+m-n$$ is

A
$$212$$

B
$$184$$

C
$$240$$

D
$$64$$

Solution

We have $$n(A)=3$$ and $$n(B)=2$$.
Now cartesian product of $$A$$ and $$B$$ can contain maximum $$3\times 2=6$$ elements.
Since relation is the possible subsets of cartesian product.
So maximum number of relations from $$A$$ to $$B$$ be $$(l)=2^{6}=64$$.
Now maximum possible symmetric relation from $$A$$ to $$A$$ be $$(m)=2^{\cfrac{3(3+1)}{2}}=2^6=64$$ and that of from $$B$$ to $$B$$ be $$(m)=2^{\cfrac{2(2+1)}{2}}=2^3=8$$. [Using formula for number of symmetric relation]
Now $$2l+m-n=128+64-8=184$$.
Question 8
1 / -0

The number of reflexive relation in set A = {a, b, c} is equal to

A
$$2^9$$

B
$$2^4$$

C
$$2^7$$

D
$$2^6$$

Solution

$$\therefore$$ Total number of relation in set A $$2 ^{3 \times 3} = 2^9$$
and maximum number of cartesian product = 9 out of which 3 ordered pair is necessary for reflexive.
So, for remaining 6 ordered pair
Number of ordered pair required.$$=^6 C _0\, +\, ^6C_1 \,+ \, ^6C_2 +......^6C_6$$ =$$ 2^6 $$
Question 9
1 / -0

If the relation is defined on $$R-\left\{ 0 \right\} $$ by $$\left( x,y \right) \in S\Leftrightarrow xy>0$$, then $$S$$ is ________

A
an equivalence relation

B
symmetric only

C
reflexive only

D
transitive only

Solution

$$s\Leftrightarrow xy$$
for (x,x) on relation
$$s\Leftrightarrow xy$$
and $$x^2>0$$
so, $$S$$ is symmetric
if $$xy>0$$
thus $$xy>0$$
so, $$S$$ is reflexive.
Now, if $$xy>0$$
and $$yz>0$$
so, $$xz>0$$
Thus, $$S$$ is transitive.
So, relation is equivilance.
Question 10
1 / -0

Let $$A=\left\{ u,v,w,z \right\} ;B=\left\{ 3,5 \right\} $$, then the number of relations from $$A$$ to $$B$$ is

A
$$256$$

B
$$1024$$

C
$$512$$

D
$$64$$

Solution

Given
$$A=\left\{ u,v,w,z \right\} ;B=\left\{ 3,5 \right\} $$
Here, number of element in set $$A$$ is $$n(A)=4$$
And number of element in set $$B$$ is $$ n(B)=2$$
$$\therefore$$ Number of relations from $$A$$ to $$B$$
$$={ 2 }^{ n(A).n(B) }={ 2 }^{ 4\times 2 }={ 2 }^{ 8 }=256$$

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Re-Attempt Weekly Quiz Competition

Relations Test 5

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Relations Test 5

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SHARING IS CARING

Question 1

$$-44$$

$$-54$$

$$-32$$

$$-64$$

Solution

Question 2

$${ 2 }^{ p+q }$$

$${ 2 }^{ pq }$$

$$p+q$$

$$pq$$

Solution

Question 3

$$4096$$

$$4094$$

$$128$$

$$126$$

Solution

Question 4

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

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

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

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

Solution

Question 5

$$2^n$$

$$n^2$$

$$2n^2$$

$$2n$$

Question 6

Reflexive, but not symmetric

Symmetric only

Reflexive and transitive

Reflexive, symmetric and transitive

Solution

Question 7

$$212$$

$$184$$

$$240$$

$$64$$

Solution

Question 8

$$2^9$$

$$2^4$$

$$2^7$$

$$2^6$$

Solution

Question 9

an equivalence relation

symmetric only

reflexive only

transitive only

Solution

Question 10

$$256$$

$$1024$$

$$512$$

$$64$$

Solution

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