Sets Test

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Sets Test - 35

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Weekly Quiz Competition

Question 1
1 / -0

If $$X=\left\{ { 4 }^{ n }-3n-1;n\in R \right\} $$ and $$Y=\left\{ 9\left( n-1 \right) ;n\in N \right\} $$, then $$X\cap Y=$$

A
$$X$$

B
$$Y$$

C
$$\phi $$

D
$$\left\{ 0 \right\} $$

Solution

Let us restrict the domain of $$X$$ to natural numbers,
$$\Longrightarrow { 4 }^{ n }-3n-1\\ \Longrightarrow { \left( 1+3 \right) }^{ n }-3n-1\\ \Longrightarrow 1+3n+{ n }_{ { C }_{ 2 } }{ 3 }^{ 2 }+...+{ 3 }^{ n }-3n-1\\ \Longrightarrow 9k$$
Where $$k$$ is some natural number,
So,the elements of $$X$$ are multiples of $$9$$,
The elements of the set $$Y$$ are also multiples of $$9$$,
But if we extend the domain of $$X$$ to the set of real numbers there would be other elements which are not present in $$Y$$ as the elements of $$Y$$ are only natural numbers where as the elements of $$X$$ can also be rational and irrational,
So the elements of $$Y$$ are contained in $$X$$ for all natural numbers,
$$\therefore X\cap Y=Y$$.
Question 2
1 / -0

The number of binary operations on the set $$\{1, 2, 3\}$$ is _________.

A
$$3^9$$

B
$$9^3$$

C
$$27$$

D
$$3!$$

Solution

Let us denote this set by $$S$$, then $$|S| = 3$$.
A binary relation defined on the elements of $$S$$ maps all elements in $$S\times S$$ to elements in $$S$$ by definition.
In this case any binary relation will thus have $$3^2 =9$$ inputs each of which is an ordered pair of elements from $$S$$ and only $$3$$ number of possible outputs.
If all possible binary operations are considered then it is possible to assign any of the $$3$$ outputs to any of the $$9$$ inputs. So the number of all binary operations would exactly be $$3^9$$.
So option A is the correct answer.
Question 3
1 / -0

Choose the correct answers from the alternatives given.
Directions for questions $$71$$ to $$75$$ : Study the pie chart carefully to answer the questions that follow.
Percentage distribution of students in different courses.
What is the value of half of the difference between the number of students in MBA and MBBS?
The Total number of student = $$6500$$

A
$$800$$

B
$$1600$$

C
$$1300$$

D
$$650$$

Solution
Question 4
1 / -0

If $$a.N = \left\{ ax\thinspace :\thinspace x\in N \right\}$$ then $$3N\cap 7N=$$

A
$$21\ N$$

B
$$10\ N$$

C
$$4\ N$$

D
$$none$$

Solution

Clearly the set $$aN$$ will include all multiples of $$a$$.

Hence $$3N$$ and $$7N$$ contain multiples of 3 and 7 respectively.

Their intersection must have the multiples of their L.C.M,i.e,21

Hence $$3N\cap 7N=21N$$
Question 5
1 / -0

The relation $$S=\{(3, 3), (4, 4)\}$$ on the set $$A=\{3, 4, 5\}$$ is __________.

A
Not reflexive but symmetric and transitive

B
Reflexive only

C
Symmetric only

D
An equivalence relation

Solution

given, relation $$S=\{(3,3),(4,4)\}$$ on the set $$A=\{3,4,5\}$$

It is not reflexive because if every element of $$A$$ is not related to itself (i.e.,) $$(5,5)$$ is absent.

$$A$$ is symmetric because if for all $$a$$ and $$b$$ in $$A$$ that $$a$$ is related to $$b$$ if and only if $$b$$ is related to $$a$$.

$$A$$ is transitive because if $$a$$ is related to $$b$$ and $$b$$ is related to $$c$$ then $$a$$ is also related to $$c$$.

Hence, not reflexive but symmetric and transitive.
Question 6
1 / -0

A set of $$n$$ numbers has the sum $$s$$. Each number of the set is increased by $$20$$, then multiplied by $$5$$, and then decreased by $$20$$. The sum of the numbers in the new set thus obtained is:

A
$$s+20n$$

B
$$5s+80n$$

C
$$s$$

D
$$5s$$

Solution

$$s={ a }_{ 1 }+{ a }_{ 2 }+....+{ a }_{ n }$$
$$s'=5({ a }_{ 1 }+20)-20+5({ a }_{ 2 }+20)-20+.....=5{ a }_{ 1 }+80+5{ a }_{ 2 }+80+....+5{ a }_{ n }+80$$
$$=5\left( { a }_{ 1 }+{ a }_{ 2 }+....+{ a }_{ n } \right) +80n=5s+80n$$
or $$\sum _{ i=1 }^{ n }{ { a }_{ i } } =s,\sum _{ i=1 }^{ n }{ { [5(a }_{ i }+20)-20] } =5\sum _{ i=1 }^{ n }{ { a }_{ i } } +\sum _{ i=1 }^{ n }{ 80 } =5s+80n\quad \quad $$
Question 7
1 / -0

If $$A= \{x:x$$ is a multiple of $$2\}, \,\,B= \{x:x$$ is a multiple of $$5\}$$ and $$C = \{x:x$$ is a multiple of 10$$\}$$, then $$ A\cap (B\cap C)$$ is equal to

A
$$A$$

B
$$B$$

C
$$C$$

D
$$\{x:x$$ is a multiple of $$100\}$$

Solution

$$A= \{x:x$$ is a multiple of $$2\}=\{2,4,6,8,10,12,14,....\}$$
$$B= \{x:x$$ is a multiple of $$5\}=\{5,10,15,20,25,....\}$$ and
$$C = \{x:x$$ is a multiple of $$10\}=\{10,20,30,40,....\}$$

Here, $$C \subset A$$ and $$C \subset B$$
$$C = A \cap B = A \cap (B \cap C)$$
$$= A \cap C = C$$
Question 8
1 / -0

If $$M = \left\{ {x:x \geqslant 7\,\,{\text{and}}\,x \in N} \right\}$$ for universal set of natural numbers, then $$M'$$ is

A
$$\left\{ {1,2,3,4,5} \right\}$$

B
$$\left\{ {1,2,3,4,5,6,7} \right\}$$

C
$$\left\{ {1,2,3,4,5,6} \right\}$$

D
$$\left\{ {0,1,2,3,4,5,6} \right\}$$

Solution

Given $$M =\{x:x\ge 7 , x\in N\}$$. $$x$$ is a natural number such that $$x\ge 7$$. Hence $$x= \{7,8,9,10,11,12,.......\infty\}$$
$$\therefore M' = N- M$$
$$=\{1,2,3,4,5,6\}$$
Question 9
1 / -0

In the equation $${ \left( x-m \right) }^{ 2 }-{ \left( x-n \right) }^{ 2 }={ \left( m-n \right) }^{ 2 }$$, $$m$$ is a fixed positive number, and $$n$$ is a fixed negative number. The set of values $$x$$ satisfying the equation is:

A
$$x\ge 0$$

B
$$x\le n$$

C
$$x=0$$

D
the set of all real numbers

E
none of these

Solution
Question 10
1 / -0

Given : A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}
Find : $$(A \, \times \, B) \, \cap \, (B \, \times \, C)$$.

A
{4,4}

B
{3,4}

C
{3,4}, {3,3}

D
{3,3}

Solution

Given $$A=\left\{ 1,2,3 \right\} $$, $$B=\left\{ 3,4 \right\} $$ and $$C=\left\{ 4,5,6 \right\} $$

1) $$A\times B=\left\{ 1,2,3 \right\}\times \left\{ 3,4 \right\} $$
$$\therefore A\times B=\left\{ \left( 1,3 \right) ,\left( 1,4 \right) ,\left( 2,3 \right) ,\left( 2,4 \right) ,\left( 3,3 \right) ,\left( 3,4 \right) \right\} $$

2) $$B\times C=\left\{ 3,4 \right\}\times \left\{ 4,5,6 \right\} $$
$$\therefore B\times C=\left\{ \left( 3,4 \right) ,\left( 3,5 \right) ,\left( 3,6 \right) ,\left( 4,4 \right) ,\left( 4,5 \right) ,\left( 4,6 \right) \right\} $$

$$\therefore \left( A\times B \right) \cap \left( B\times C \right) =\left\{ \left( 1,3 \right) ,\left( 1,4 \right) ,\left( 2,3 \right) ,\left( 2,4 \right) ,\left( 3,3 \right) ,\left( 3,4 \right) \right\} \cap \left\{ \left( 3,4 \right) ,\left( 3,5 \right) ,\left( 3,6 \right) ,\left( 4,4 \right) ,\left( 4,5 \right) ,\left( 4,6 \right) \right\} $$

$$\therefore \left( A\times B \right) \cap \left( B\times C \right) =\left\{ \left( 3,4 \right) \right\} $$

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Re-Attempt Weekly Quiz Competition

Sets Test - 35

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Sets Test - 35

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SHARING IS CARING

Question 1

$$X$$

$$Y$$

$$\phi $$

$$\left\{ 0 \right\} $$

Solution

Question 2

$$3^9$$

$$9^3$$

$$27$$

$$3!$$

Solution

Question 3

$$800$$

$$1600$$

$$1300$$

$$650$$

Solution

Question 4

$$21\ N$$

$$10\ N$$

$$4\ N$$

$$none$$

Solution

Question 5

Not reflexive but symmetric and transitive

Reflexive only

Symmetric only

An equivalence relation

Solution

Question 6

$$s+20n$$

$$5s+80n$$

$$s$$

$$5s$$

Solution

Question 7

$$A$$

$$B$$

$$C$$

$$\{x:x$$ is a multiple of $$100\}$$

Solution

Question 8

$$\left\{ {1,2,3,4,5} \right\}$$

$$\left\{ {1,2,3,4,5,6,7} \right\}$$

$$\left\{ {1,2,3,4,5,6} \right\}$$

$$\left\{ {0,1,2,3,4,5,6} \right\}$$

Solution

Question 9

$$x\ge 0$$

$$x\le n$$

$$x=0$$

the set of all real numbers

none of these

Solution

Question 10

{4,4}

{3,4}

{3,4}, {3,3}

{3,3}

Solution

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