Set Theory Test 22

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Set Theory Test 22

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Weekly Quiz Competition

Question 1
1 / -0

If X and Y are two sets such that n(X)=17, n(Y)=23 and n(X $$\cup$$ Y)=38, find n(X $$\cap$$ Y).

A
$$2$$

B
$$4$$

C
$$6$$

D
$$8$$

Solution

$$\textbf{Step 1: Label the given information and use relevant formula}$$
$${\text{It is given that n(X) = 17,n(Y) = 23,n(X}} \cup {\text{Y) = 38}}$$
$${\text{As we know that, n(X}} \cup {\text{Y) = n(X) + n(Y) - n(X}} \cap {\text{Y)}}$$
$${\text{Using the formula we get}}$$
  $${\text{n(X) + n(Y) - n(X}} \cap {\text{Y) = (X}} \cup {\text{Y)}}$$
  $$\therefore {\text{17 + 23 - n(X}} \cap {\text{Y) = 38}}$$
  $$ \Rightarrow {\text{n(X}} \cap {\text{Y) = 40 - 38 = 2}}$$
$${\textbf{Hence, the value is 2.}}$$
Question 2
1 / -0

If $$X$$ and $$Y$$ are two sets, then $$X\cap \left( X\cup Y \right)$$ equals

A
$$X$$

B
$$Y$$

C
$$\phi$$

D
$$None\ of\ these$$

Solution

Ans. $$(a).$$
If $$X\subset Y$$, then $$X\cup Y=Y$$ and $$X\cap Y=X$$
If $$Y\subset X$$, then $$X\cup Y=X$$ and $$X\cap X=X$$
Question 3
1 / -0

If a relation from a finite set A having m elements to a finite set B having n elements, then the number of relations from A to B is

A
$$2^{mn}$$

B
$$2^{mn}-1$$

C
2 mn

D
$$m^n$$
Question 4
1 / -0

Let $$A=\left\{ x:x\ \in\ R,\left| x \right| <1 \right\}$$
$$B=\left\{ x:x\ \in\ R,\left| x-1 \right| \ge 1 \right\}$$
and $$A\cup B=R-D$$, then set $$D$$ is

A
$$\left\{ x:1 < x \le 2 \right\}$$

B
$$\left\{ x:1\le x<2 \right\}$$

C
$$\left\{ x:1\le x\le 2 \right\}$$

D
None of these

Solution

Given $$A=\left\{ x:x\in R\left| x \right| <1 \right\} $$

If $$x$$ is positive, $$\left| x \right| =x$$
$$\therefore x<1$$

If $$x$$ is negative, $$\left| x \right| =-x$$
$$\therefore -x<1$$
$$\therefore x>-1$$

$$\therefore A=\left\{ -1<x<1 \right\} $$ (1)

Given $$B=\left\{ x:x\in R,\left| x-1 \right| \ge 1 \right\} $$

If $$\left| x-1 \right| $$ is positive, $$\left| x-1 \right| =x-1$$
$$\therefore x-1\ge 1$$
$$\therefore x\ge 2$$

If $$\left| x-1 \right| $$ is negative, $$\left| x-1 \right| =-\left( x-1 \right) $$
$$\therefore -\left( x-1 \right) \ge 1$$
$$\therefore \left( x-1 \right) \le -1$$
$$\therefore x\le 0$$

$$\therefore B=\left\{ x\le 0\quad and\quad x\ge 2 \right\} $$ (2)

Thus, $$A\cup B=\left\{ -1<x<1 \right\} \cup \left\{ x\le 0\quad and\quad x\ge 2 \right\}$$

The region which is not included in this domain has to be subtracted from remaining domain of real numbers.

$$\therefore A\cup B=R-D$$

Thus, set D is set of excluded region which is the region between 1 and 2 including 1.

$$\therefore D:-1\le x<2$$
Question 5
1 / -0

Consider the word $$W=MISSISSIPPI$$.
If $$N$$ denotes the number of different selections of $$5$$ letters from the word $$W = MISSISSIPPI$$ then $$N$$ belongs to the set,

A
$$\{ 15,\ 16,\ 17,\ 18,\ 19\} $$

B
$$\{ 20,\ 21,\ 22,\ 23,\ 24\} $$

C
$$\{ 25,\ 26,\ 27,\ 28,\ 29\} $$

D
$$\{ 30,\ 31,\ 32,\ 33,\ 34\} $$

Solution

In $$MISSISSIPPI$$
$$M=1\quad I=4\quad S=4\quad P=2$$
$$5$$ letters word can be made as:-
$$^{2}C_{1} \ ^{3}C_{1}+ ^{2}C_{1} + ^{2}C_{1} ^{3}C_{2} + ^{3}C_{1} \times 1$$
$$=6+4+6+6+3$$
$$=23$$
the set is $$\left\{20, 21, 22, 23, 24\right\}$$
Question 6
1 / -0

If sets $$A$$ and $$B$$ are define as
$$A=\left\{ \left( x,y \right) :y={ e }^{ x },x\in R \right\}$$
$$B=\left\{ \left( x,y \right) :y=x,x\in R \right\}$$, then

A
$$B\subset A$$

B
$$A\subset B$$

C
$$A\cap B=\phi$$

D
$$A\cup B=A$$

Solution

Given $$A=\left\{ \left( x,y \right) :y={ e }^{ x },x\in R \right\} $$

Thus, in roaster form, set A can be written as,
$$A=\left\{ .............\left( -4,\frac { 1 }{ { e }^{ 4 } } \right) ,\left( -3,\frac { 1 }{ { e }^{ 3 } } \right) ,\left( -2,\frac { 1 }{ { e }^{ 2 } } \right) ,\left( -1,\frac { 1 }{ { e }^{ 1 } } \right) ,\left( 0,1 \right) ,\left( 1,{ e }^{ 1 } \right) ,\left( 2,{ e }^{ 2 } \right) ,\left( 3,{ e }^{ 3 } \right) ,\left( 4,{ e }^{ 4 } \right) ........... \right\} $$ (1)

Given $$B=\left\{ \left( x,y \right) :y=x,x\in R \right\}$$

Thus, in roaster form, set B can be written as,
$$B=\left\{ ...........\left( -4,-4 \right) ,\left( -3,-3 \right) ,\left( -2,-2 \right) ,\left( -1,-1 \right) ,\left( 0,0 \right) ,\left( 1,1 \right) ,\left( 2,2 \right) ,\left( 3,3 \right) ,\left( 4,4 \right) ......... \right\} $$ (2)

From equations (1) and (2), it is clear that no element is common in both the sets.

$$\therefore A\cap B=\phi $$
Question 7
1 / -0

Let $$A=\left\{ \left( x,y \right) :y={ e }^{ x },x\in R \right\}$$
$$B=\left\{ \left( x,y \right) :y={ e }^{ -x },x\in R \right\}$$. Then

A
$$A\cap B=\phi$$

B
$$A\cap B\neq \phi$$

C
$$A\cup B={ R }^{ 2 }$$

D
$$none\ of\ these$$

Solution

Ans. $$(b)$$.
$$y={e}^{x},\ y={e}^{-x},$$ meet where $${ e }^{ x }=\ { e }^{ -x }$$ or $$e^{2x}=1$$
$$\therefore x=0$$ and hence $$y=1$$. These curve meet at $$(0,1)$$ so that $$A\cap B=(0,1)$$.
In other words $$A\cap B\neq \phi $$.
Question 8
1 / -0

If $$X$$ and $$Y$$ are two sets, then $$X\cap \left( Y\cup X \right)'$$ equals

A
$$X$$

B
$$Y$$

C
$$\phi$$

D
$$None\ of\ these$$

Solution

$$X\cap \left( X' \cap Y' \right) =\left( X\cap X' \right) \cap Y' =\phi \cap Y' =\phi$$
Question 9
1 / -0

Suppose $${A_1},{A_2},...,{A_{30}}$$ are thirty sets each having 5 elements and $${B_1},{B_2},...,{B_n}$$ are $$n$$ sets each with 3 elements, let $$\bigcup\limits_{i = 1}^{30} {{A_i}} = \bigcup\limits_{i = 1}^n {Bj} = S$$ and each element of $$S$$ belongs to exactly 10 of the $${A_i}'s$$ and exactly 9 of the $$B,'s$$. Then $$n$$ is equal to

A
$$15$$

B
$$3$$

C
$$45$$

D
$$35$$

Solution
Question 10
1 / -0

In certain town, $$25\%$$ families own a cell phone, $$15\%$$ families own a scooter and $$65\%$$ families own neither a cell phone nor a scooter. If $$1500$$ families own both a cell phone and a scooter, then the total number of families in the town is:

A
$$10000$$

B
$$20000$$

C
$$30000$$

D
$$50000$$

Solution

65% of families own neither a cell phone nor a scooter
35% of families own either a phone or a scooter
(A$$\cup$$B)=A+B-(A$$\cap$$B)
35%x=25%x+15%x-(A$$\cap$$B)
(A$$\cap$$B)=5%x=$$\dfrac{5}{100}x=1500$$
number of families=30000

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Set Theory Test 22

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Set Theory Test 22

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Question 1

$$2$$

$$4$$

$$6$$

$$8$$

Solution

Question 2

$$X$$

$$Y$$

$$\phi$$

$$None\ of\ these$$

Solution

Question 3

$$2^{mn}$$

$$2^{mn}-1$$

2 mn

$$m^n$$

Question 4

$$\left\{ x:1 < x \le 2 \right\}$$

$$\left\{ x:1\le x<2 \right\}$$

$$\left\{ x:1\le x\le 2 \right\}$$

None of these

Solution

Question 5

$$\{ 15,\ 16,\ 17,\ 18,\ 19\} $$

$$\{ 20,\ 21,\ 22,\ 23,\ 24\} $$

$$\{ 25,\ 26,\ 27,\ 28,\ 29\} $$

$$\{ 30,\ 31,\ 32,\ 33,\ 34\} $$

Solution

Question 6

$$B\subset A$$

$$A\subset B$$

$$A\cap B=\phi$$

$$A\cup B=A$$

Solution

Question 7

$$A\cap B=\phi$$

$$A\cap B\neq \phi$$

$$A\cup B={ R }^{ 2 }$$

$$none\ of\ these$$

Solution

Question 8

$$X$$

$$Y$$

$$\phi$$

$$None\ of\ these$$

Solution

Question 9

$$15$$

$$3$$

$$45$$

$$35$$

Solution

Question 10

$$10000$$

$$20000$$

$$30000$$

$$50000$$

Solution

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