Set Theory Test 5

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Set Theory Test 5

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Question 1
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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,

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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 2
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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 3
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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 4
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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 5
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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 $$A\cap B=A$$.
Question 6
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Let $$A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$$. Then the number of subsets of $$A$$ containing exactly two elements is

A
$$20$$

B
$$40$$

C
$$45$$

D
$$90$$

Solution

Number of elements in $$A= 10$$
Number of subsets of $$A$$ containing exactly two elements
$$=$$ Number of ways we can select $$ 2 $$ elements from $$10$$ elements
$${}^{ 10 }{ C }_{2}=\dfrac { 10\times 9 }{ 2 } =45$$
$$\therefore$$ Number of subsets of $$A$$ containing exactly two elements $$=45$$
Question 7
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In a class of $$60$$ students, $$45$$ students like music, $$50$$ students like dancing, $$5$$ students like neither. Then the number of students in the class who like both music and dancing is

A
$$35$$

B
$$40$$

C
$$50$$

D
$$55$$

Solution

Total number of students in class $$=60$$
Students who like music $$=X+Y=45$$
Students who like dancing $$=Y+Z=50$$
Students who like nothing $$=5$$
Students who like both music and dancing $$=Y$$
We know that $$X+Y+Z+5=60$$
$$\Rightarrow 45+Z+5=60$$
$$\Rightarrow Z=10$$
$$\Rightarrow Y=40$$
Hence, $$40$$ students like both music and dancing.
Question 8
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Let $$A_1, A_2$$ and $$A_3$$ be subsets of a set $$X$$. Which one of the following is correct?

A
$$A_1\cup{A_2}\cup{A_3}$$ is the largest subset of $$X$$ containing elements of each of $$A_1, A_2$$ and $$A_3$$

B
$$A_1\cup{A_2}\cup{A_3}$$ is the smallest subset of $$X$$ containing either $$A_1$$ or $$A_2\cup{A_3}$$ but not both

C
The smallest subset of $$X$$ containing $$A_1\cup{A_2}$$ and $$A_3$$ equals the smallest subset of $$X$$ containing both $$A_1$$ and $$A_2\cup{A_3}$$ only if $$A_2=A_3$$

D
None of these

Solution

Since union of subsets is the smallest subset containing all the elements of the subsets.
Hence, option (A) is correct.
Question 9
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Statements:
I. Some Politicians are social workers.
II. All Doctors are social workers.
Conclusions:
I. Some Doctors are Politicians.
II. Some social workers are Doctors as well as Politicians.

A
Both conclusion (I) and (II) follow

B
Only conclusion (II) follow

C
Neither conclusion (I) nor (II) follow

D
Only conclusion (I) follow

Solution

R.E.F image
Thus, option $$(3)$$ is correct.
Question 10
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From among the given alternatives select the one in which the set of numbers is most like the set of numbers given in the question.
Given set $$:$$ $$(7, 15, 31)$$

A
$$7, 13, 28$$

B
$$5, 13, 28$$

C
$$9, 13, 26$$

D
$$5, 13, 29$$

Solution

Let us find the Relation between the numbers of the set $$(7 , 15 , 31)$$.
$$(7)\times 2 +1 = (15)$$
$$(15)\times 2 +1 = (31)$$

Options $$A,B,C$$ are not of similar type of above set

This similar type of relation is shown by option D $$(7 , 15 , 31)$$.
$$(5)\times 2 +3 = (13)$$
$$(13)\times 2 +3 = (29)$$

Hence, option D is correct.

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Re-Attempt Weekly Quiz Competition

Set Theory Test 5

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Set Theory Test 5

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SHARING IS CARING

Question 1

$$n(P) = 45$$

$$n(P) = 40$$

$$n(P) = 50$$

$$n(P) = 55$$

Solution

Question 2

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

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

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

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

Solution

Question 3

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 4

$$3$$

$$2$$

$$1$$

$$0$$

Solution

Question 5

$$B$$

$$A\setminus B$$

$$A$$

$$B \setminus A$$

Solution

Question 6

$$20$$

$$40$$

$$45$$

$$90$$

Solution

Question 7

$$35$$

$$40$$

$$50$$

$$55$$

Solution

Question 8

$$A_1\cup{A_2}\cup{A_3}$$ is the largest subset of $$X$$ containing elements of each of $$A_1, A_2$$ and $$A_3$$

$$A_1\cup{A_2}\cup{A_3}$$ is the smallest subset of $$X$$ containing either $$A_1$$ or $$A_2\cup{A_3}$$ but not both

The smallest subset of $$X$$ containing $$A_1\cup{A_2}$$ and $$A_3$$ equals the smallest subset of $$X$$ containing both $$A_1$$ and $$A_2\cup{A_3}$$ only if $$A_2=A_3$$

None of these

Solution

Question 9

Both conclusion (I) and (II) follow

Only conclusion (II) follow

Neither conclusion (I) nor (II) follow

Only conclusion (I) follow

Solution

Question 10

$$7, 13, 28$$

$$5, 13, 28$$

$$9, 13, 26$$

$$5, 13, 29$$

Solution

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