Sets Test

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Sets Test - 14

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Weekly Quiz Competition

Question 1
1 / -0

Let $$P =$$ Set of all integral multiples of $$3 $$; $$Q =$$ Set of integral multiples of $$4 $$; $$R =$$ Set of all integral multiples of $$6$$. Consider the following relations :
$$1 $$ $$\displaystyle P\cup Q=R$$
$$2.$$ $$\displaystyle P\subset R$$
$$3.$$ $$\displaystyle R\subset \left ( P\cup Q \right )$$
Which of the relations given above is/are correct ?

A
only $$1$$

B
only $$2$$

C
only $$3$$

D
both $$2$$ and $$3$$

Solution

Given that P =$$\{3, 6, 9, 12, 15, 18, 21,...\}$$
Q = $$\{4, 8, 12, 16, 20,...\}$$
R = $$\{6, 12, 18,...\}$$
Considering the choices given :
1. P $$\displaystyle \cup $$ Q = $$\{3, 4, 6, 8, 9, 12, 16, 18,...\}$$ $$\displaystyle \neq $$ R
2. All the element of P are not in R so P $$\displaystyle \nsubseteq $$ R
3. P $$\displaystyle \cup $$ Q = $$\{3, 4, 6, 8, 9, 12, 15, 16, 18, ...\}$$
$$\displaystyle \Rightarrow $$ All the element of R are in P $$\displaystyle \cup $$ Q
$$\displaystyle \Rightarrow $$R $$\displaystyle \subset $$ (P$$\displaystyle \cup $$ Q)
Question 2
1 / -0

In a locality two-thirds of the people have cable TV one-fifth have Dish TV and one-tenth have both What is the fraction of people having either cable TV or Dish TV?

A
$$\displaystyle \frac{2}{3}$$

B
$$\dfrac13$$

C
$$\dfrac45$$

D
$$\dfrac35$$

Solution

Fraction of people who watch cable only $$\displaystyle =\left ( \frac{2}{3}-\frac{1}{10} \right )=\frac{17}{30}$$
Fraction of people who watch Dish TV only $$\displaystyle =\left ( \frac{1}{5}-\frac{1}{10} \right )=\frac{1}{10}$$
$$\displaystyle\therefore $$ Fraction of people who watch either cable of Dish TV $$\displaystyle =\frac{17}{30}+\frac{1}{10}=\frac{20}{30}=\frac{2}{3}$$
Question 3
1 / -0

In an examination $$70\%$$ students passed both in Mathematics and Physics $$85\%$$ passed in Mathematics and $$80\%$$ passed in Physics If $$30$$ students have failed in both the subjects then the total number of students who appeared in the examination is equal to :

A
$$900$$

B
$$600$$

C
$$150$$

D
$$100$$

Solution

Student passed in atleast one subject
$$= n$$ $$\displaystyle \left ( P\cup M \right )= n\left ( P \right )+n\left ( M \right )-n\left ( P\cup M \right )$$
$$= 80 + 85 - 70 = 95$$
$$\displaystyle \therefore $$ $$5\%$$ student failed in both the subjects
$$\displaystyle \Rightarrow $$$$5\%$$ of total students $$= 30$$
$$\displaystyle \Rightarrow $$Total students = $$\displaystyle \frac{30\times 100}{5}= 600$$
Question 4
1 / -0

In a class of $$50$$ students $$35$$ opted for Mathematics and $$37$$ opted for Biology How may have opted for only Mathematics? ( Assume that each student has to opt for at least one of the subjects)

A
$$15$$

B
$$17$$

C
$$13$$

D
$$19$$

Solution

Here $$n$$$$\displaystyle \left ( M\cup B \right )$$ $$= 50$$, $$n(M) = 35$$, $$n(B) = 37$$
$$\displaystyle \therefore n\left ( M\cap B \right )=n\left ( M \right )+n(B)-n\left ( M\cup B \right )$$
$$= 35 + 37 - 50 = 22$$
$$\displaystyle \Rightarrow $$ $$22 $$ student have opted for both Mathematics and Biology.
Now the number of students who have opted for Mathematics only
$$= n(M) - n$$$$\displaystyle \left ( M\cap B \right )$$
$$= 35 - 22 = 13$$
Question 5
1 / -0

If $$n(A) = 65, n(B) = 32$$ and $$\displaystyle n\left ( A\cap B \right )=14 $$, then $$\displaystyle n\left ( A\Delta B \right ) $$ equals

A
$$65$$

B
$$47$$

C
$$97$$

D
$$69$$

Solution

$$\displaystyle n(A\Delta B)=n (A-B)+n(B-A)$$
$$\displaystyle \therefore A\Delta B=(A-B)\cup (B-A)$$
$$\Rightarrow A\Delta B=\displaystyle n(A)-n(A\cap B)+n(B)-n(A\cap B)$$
$$\Rightarrow A\Delta B =n(A)+n(B)-2n(A\cap B)=65+32-2\times 14=69$$
Hence, $$A\Delta B=69$$
Question 6
1 / -0

If $$n(A) = 115$$, $$n(B) = 326$$, $$n(A - B) = 47$$ then $$\displaystyle n\left ( A\cup B \right )$$ is equal to

A
$$373$$

B
$$165$$

C
$$370$$

D
None

Solution

$$n(A)=115,n(B)=326$$
$$n(A-B)=47$$
$$n(A)=n(A-B)+n(A\cap B)$$
$$n(A\cap B)=n(A)-n(A-B)$$
$$\therefore n(A\cap B)=115-47=68$$
$$\therefore n(A\cup B)=n(A)+n(B)-n(A\cap B)$$
$$\Rightarrow n(A\cup B)=115+326-68$$
$$\Rightarrow n(A\cup B)=373$$
Question 7
1 / -0

If $$X$$ and $$Y$$ are any two non empty sets then what is $$\displaystyle \left ( X-Y \right )'$$ equal to?

A
$$X' - Y'$$

B
$$\displaystyle {X}'\cap Y $$

C
$$\displaystyle {X}'\cup Y $$

D
$$X - Y'$$

Solution

$$X - Y$$ $$\displaystyle =\left \{ x :x \in X,x\notin Y \right \}$$
$$\displaystyle =\left \{ x :x \in X,x\in Y'\right \}$$
$$\displaystyle \Rightarrow \left \{ x : x\in X\cap Y' \right \}$$
$$\displaystyle \Rightarrow \left ( X-Y \right )'=(X\cap Y)'$$
= $$\displaystyle X'\cup (Y')=X'\cup Y$$
Question 8
1 / -0

If $$A$$ and $$B$$ are non empty sets and A' and B' represents their compliments respectively then

A
$$A - B = A' - B'$$

B
$$A - A' = B - B'$$

C
$$A - B = B' - A'$$

D
$$A - B' = A' - B$$

Solution

Let $$ U \to$$ Universal set
$$X\to U - (A+B) $$
$$B'=X+A$$
$$A'=X+B$$
$$B'-A'= X+A-(X+B)$$
$$=X+A-X-B$$
$$B'-A'=A-B$$
Question 9
1 / -0

If $$\displaystyle \xi =\left \{ 2,3,4,5,6,7,8,9,10,11 \right \}$$
$$\displaystyle A =\left \{ 3,5,7,9,11 \right \}$$
$$\displaystyle B =\left \{ 7,8,9,10,11 \right \}$$, then find $$(A - B)'$$

A
$$(A - B)' = \{2,4,6,7,8,9,10,11\}$$

B
$$(A - B)' = \{2,4,6,7,8,9,11\}$$

C
$$(A - B)' = \{2,4,6,7,9,10,11\}$$

D
$$(A - B)' = \{2,4,6,8,9,10,11\}$$

Solution

$$A - B = \{3, 5\}$$
$$(A - B)' = \{2,4,6,7,8,9,10,11\}$$
Question 10
1 / -0

If $$\displaystyle Q=\left\{ x:x=\frac { 1 }{ y } ,where\ \ y\ \in \ N \right\} $$, then find the correct one.

A
$$\displaystyle 0\quad \in \quad Q$$

B
$$\displaystyle 1\quad \in \quad Q$$

C
$$\displaystyle 2\quad \in \quad Q$$

D
$$\displaystyle \frac { 2 }{ 3 } \quad \in \quad Q$$

Solution

(A) Since $$\displaystyle y\neq\frac{1}{0}$$ then $$0\notin Q$$

(B) Since $$\displaystyle y=\frac{1}{2}$$ as $$1\in N$$ then $$1\in Q$$

(C) Since $$\displaystyle y\neq\frac{1}{2}$$ as $$\displaystyle\frac{1}{2}\notin N$$ then $$2\notin Q$$

(D) Since $$\displaystyle y\neq\frac{3}{2}$$ as $$\displaystyle\frac{3}{2}\notin N$$ then $$\displaystyle\frac{2}{3}\notin Q$$

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Re-Attempt Weekly Quiz Competition

Sets Test - 14

Result

Sets Test - 14

Score

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Rank

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SHARING IS CARING

Question 1

only $$1$$

only $$2$$

only $$3$$

both $$2$$ and $$3$$

Solution

Question 2

$$\displaystyle \frac{2}{3}$$

$$\dfrac13$$

$$\dfrac45$$

$$\dfrac35$$

Solution

Question 3

$$900$$

$$600$$

$$150$$

$$100$$

Solution

Question 4

$$15$$

$$17$$

$$13$$

$$19$$

Solution

Question 5

$$65$$

$$47$$

$$97$$

$$69$$

Solution

Question 6

$$373$$

$$165$$

$$370$$

None

Solution

Question 7

$$X' - Y'$$

$$\displaystyle {X}'\cap Y $$

$$\displaystyle {X}'\cup Y $$

$$X - Y'$$

Solution

Question 8

$$A - B = A' - B'$$

$$A - A' = B - B'$$

$$A - B = B' - A'$$

$$A - B' = A' - B$$

Solution

Question 9

$$(A - B)' = \{2,4,6,7,8,9,10,11\}$$

$$(A - B)' = \{2,4,6,7,8,9,11\}$$

$$(A - B)' = \{2,4,6,7,9,10,11\}$$

$$(A - B)' = \{2,4,6,8,9,10,11\}$$

Solution

Question 10

$$\displaystyle 0\quad \in \quad Q$$

$$\displaystyle 1\quad \in \quad Q$$

$$\displaystyle 2\quad \in \quad Q$$

$$\displaystyle \frac { 2 }{ 3 } \quad \in \quad Q$$

Solution

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