Sets Test

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Sets Test - 17

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Weekly Quiz Competition

Question 1
1 / -0

A survey was carried out to find out the types of shampoo that a group of $$150$$ women have tried. It was found that $$84$$ women have used brand A shampoo, $$93$$ have used brand B, and $$69$$ have used brand C of these women, $$45$$ have tried brands A and B, $$25$$ have tried brands A and C and $$40$$ have tried brand B and C. Determine the number of women who have tried (a) all three brands, (b) only brand A

A
(a) $$14$$ (b) $$28$$

B
(a) $$15$$ (b) $$28$$

C
(a) $$14$$ (b) $$29$$

D
(a) $$16$$ (b) $$29$$

Solution

$$n(A\cup B\cup C)=150$$
$$n(A)=84$$
$$n(B)=93$$
$$n(C)=69$$
$$n(A\cap C)=25$$
$$n(A\cap B)=45$$
$$n(B\cap C)=40$$
$$(a)$$ The number of women who have tried all three brands
$$n(A\cap B\cap C)=n(A\cup B\cup C)-n(A)-n(B)-n(C)+n(A\cap B)+n(A\cap C)+N(B\cap C)$$
$$=150-84-93-69+45+25+40$$
$$=14$$

$$(b)$$ The number of women who have tried only brand $$A$$
The number of women who have tried only brand $$A$$ $$=n(A)-\left[n(A\cap B)+ n(A\cap C)\right]+n(A\cap B\cap C)$$
$$=84-[45+25]+14$$
$$=84-70+14$$
$$=28$$
Question 2
1 / -0

If $$A = \left \{1, 2, 3, 4\right \}$$, what is the number of subsets of A with at least three elements?

A
$$3$$

B
$$4$$

C
$$5$$

D
$$10$$

Solution

A subset containing $$3$$ elements $$= \left \{1, 2, 3\right \}; \left \{1, 3, 4\right \}; \left \{1, 2, 4\right \}$$ and $$\left \{2, 3, 4\right \}$$
A subset containing $$4$$ elements $$= \left \{1, 2, 3, 4\right \}$$
$$\therefore$$ there are five subsets containing at least $$3$$ elements.
Question 3
1 / -0

The set of integers is closed with respect to which one of the following?

A
Addition only

B
Multiplication only

C
By addition and multiplication

D
Division

Solution

From group theory, integers are closed w.r.t. both addition & multiplication
Question 4
1 / -0

In a survey, it was fond that $$65$$% of the people watched news on TV, $$40$$% read in newspaper, $$25$$% read newspaper and watched TV. What percentage of people neither watched TV nor read newspaper?

A
$$0$$%

B
$$5$$%

C
$$10$$%

D
$$20$$%

Solution

n(Watch new on TV) $$=65\%=n(T)$$

n(Reads newspaper) $$=40\%=n(N)$$

n(Watch news on TV & read newspapers ) $$=25\%=n(T\cap N)$$

$$n(N \cup T)= n(T)+n(N)-n(N\cap T)$$

$$n(N \cup T)= 65+40-25 = 80\%$$
$$n(N\cap T)^{c}=100-80=20\%$$
Question 5
1 / -0

In the Venn diagram, the numbers represent the number of elements in the subsets. Given that $$\xi = F\cup G\cup H$$ and $$n(\xi) = 42$$, find $$n(G'\cup H)$$

A
$$18$$

B
$$28$$

C
$$30$$

D
$$38$$

Solution

$$n(\xi) = 42$$

$$n(G) = 12, n(F) = 17, n(H) = 8$$

$$n(F \cap G) = 5, n(G \cap H) = 0, n(F \cap H) = 0$$

$$n(G’)= 42-7-5 =30$$

$$n(H)= 8, n(G’ \cup H) = n(G')+n(H)-n(G'\cap H)=30+8-8 = 30$$
Question 6
1 / -0

Directions For Questions

In a certain group of $$72$$ students are taking physics, geography and english; $$24$$ students are taking physics and geography, $$30$$ students are taking physics and english; and $$22$$ students are taking geography and english. However, $$7$$ students are taking only physics, $$10$$ students are taking only geography and $$5$$ students are taking only english,

...view full instructions

How many of these students are taking physics?

A
$$n(P) = 45$$

B
$$n(P) = 40$$

C
$$n(P) = 50$$

D
$$n(P) = 55$$

Solution

$$n(P\cap G\cap E)=16$$
$$n(P\cap G)=24$$
$$n(P\cap E)=30$$
$$n(G\cap E)=22$$
Only $$7$$ students are taking Physics.
$$\therefore$$ $$7=n(P)-[n(P\cap G)+n(P\cap E)]+n(P\cap G\cap E)$$

$$\Rightarrow$$ $$7=n(P)-[24+30]+16$$

$$\Rightarrow$$ $$7=n(P)-54+16$$

$$\Rightarrow$$ $$n(P)=7+54-16$$

$$\therefore$$ $$n(P)=45$$
Question 7
1 / -0

Which one of the following is incorrect?

A
Every subset of a finite set is finite

B
$$P=\left \{ x:x-8=-8 \right \}$$ is a singleton set

C
Every set has a proper set

D
Every non-empty set has at least two subsets, $$\phi$$ and the set itself

Solution

The void set has only one subset which is equal to itself. Hence it has no proper subset.
Question 8
1 / -0

The number of elements of the set $$\left \{ x:x\in Z,x^{2}=1 \right \}$$ is :

A
$$3$$

B
$$2$$

C
$$1$$

D
$$0$$

Solution

$$x^2=1\Rightarrow x=1,-1$$

Since both solutions are integers the set has $$2$$ elements
Question 9
1 / -0

Upper limit of class $$'41 - 50'$$ is __________.

A
$$41$$

B
$$50$$

C
$$45$$

D
$$91$$

Solution

Upper Limit of class $$'41-50'=50$$
Part (B) is correct answer.
Question 10
1 / -0

If $$A \subset B$$, then $$A \cap B$$ is

A
$$B$$

B
$$A\setminus B$$

C
$$A$$

D
$$B \setminus A$$

Solution

We are given that $$A$$ is the subset of $$B$$
$$\Rightarrow$$ Every element of $$A$$ is an element of $$B$$.
Therefore, the intersection elements of sets $$A$$ and $$B$$ are $$A\cap B=A$$.

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Re-Attempt Weekly Quiz Competition

Sets Test - 17

Result

Sets Test - 17

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SHARING IS CARING

Question 1

(a) $$14$$ (b) $$28$$

(a) $$15$$ (b) $$28$$

(a) $$14$$ (b) $$29$$

(a) $$16$$ (b) $$29$$

Solution

Question 2

$$3$$

$$4$$

$$5$$

$$10$$

Solution

Question 3

Addition only

Multiplication only

By addition and multiplication

Division

Solution

Question 4

$$0$$%

$$5$$%

$$10$$%

$$20$$%

Solution

Question 5

$$18$$

$$28$$

$$30$$

$$38$$

Solution

Question 6

$$n(P) = 45$$

$$n(P) = 40$$

$$n(P) = 50$$

$$n(P) = 55$$

Solution

Question 7

Every subset of a finite set is finite

$$P=\left \{ x:x-8=-8 \right \}$$ is a singleton set

Every set has a proper set

Every non-empty set has at least two subsets, $$\phi$$ and the set itself

Solution

Question 8

$$3$$

$$2$$

$$1$$

$$0$$

Solution

Question 9

$$41$$

$$50$$

$$45$$

$$91$$

Solution

Question 10

$$B$$

$$A\setminus B$$

$$A$$

$$B \setminus A$$

Solution

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