Question 1: Let A = \{ a, b, c, e, f \} and B = \{ c, d, e, g \} be the two subset of the universal set x = \{ a, b, c, d, e, f, g, h \} . Draw the Venn diagrams to represent these sets. From the Venn diagrams so drawn, find:   (i) A \cap B      (ii) A'     (iii) B'    (iv) A \cup B

Answer:

V1

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Question 2. From the Adjoining figure, find:V2

(i) A \cap B      (ii) A \cup B      (iii) A'

(iv) B'      (v) (A \cup B)'       (vi) (A \cap B)'

Answer:

(i)  A \cap B = \{10, 13 \}

Note: Take the common elements between A and B

(ii) A \cup B = \{10, 11, 12, 13, 16, 17, 19 \}

Note: Take all the elements from the universal set which are not in A and B

(iii)  A' = \{11, 12, 14, 15, 17, 18 \}

Note: All elements that are in the universal set but not in A

(iv)  B' = \{14, 15, 16, 17, 18, 19 \}

Note: All elements that are in the universal set but not in B

(v)  (A \cup B)' = \{14, 15, 17, 18 \}

Note: All elements that are in the universal set but not in (A \cup B)

(vi)  (A \cap B)' = \{11, 12, 14, 15, 16, 17, 18, 19 \}

Note: All elements that are in the universal set but not in (A \cap B)

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Question 3: Use the adjoining figure, find the following:V3

(i)    A \cap B      (ii)    A \cap C      (iii)    B \cap C      (iv)    A \cap B \cap C      (v )   C'      (vi)    A \cup B      (vii)    A - B       (viii)    B - C       (ix)    C - A       (x)    (B \cup C)'      (xi)    (A \cup C)'      (xii)    (A \cup B \cup C)'

Answers:

(i)    A \cap B = \{2, 4 \}      (ii)    A \cap C = \{4, 5 \}      (iii)    B \cap C = \{4, 6 \}

(iv)    A \cap B \cap C = \{4 \}      (v)    C' = \{2, 3, 9, 10 11 \}    (vi)    A \cup B = \{2, 3, 4, 5, 6, 10 \}

(vii)    A - B = \{5, 10 \}      (vii)  B - C = \{2, 3 \}      (ix)    C - A = \{6, 7, 8 \}

(x)  (B \cup C)' = \{9, 10, 11 \}    (xi)    (A \cup C)' = \{3, 9, 10 \}      (xii)  (A \cup B \cup C)' = \{9, 11 \}

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Question 4. Use the adjacent Venn diagram to find:V4

(i) A \cap B      (ii) A \cup B      (iii) B - A      (iv) A - B      (v) A     (vi) A'

Answers:

(i) A \cap B = {2, 4}      (ii) A \cup B = {2, 4, 5, 7, 9}

(iii) B - A = \phi

Note: There are no elements in B which are not in A . Hence Null set.

(iv) A - B = {5, 7, 9}      (v) A = {2, 4, 5, 7, 9}      (vi) A' = {1, 3, 6, 8, 10}

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Question 5: Let A and B be sets such that n(A) = 17 , n(A \cup B)=38 and n(A \cap B) = 2 . Draw Venn diagrams and find:V5

(i) n(A - B)      (ii) n(B)      (iii) n(B - A)

Answers:

(i) n(A - B) = 13

Note: Elements which are in A but not in B (15-2)

(ii) n(B) = 23

Note: Total number of elements in B which are (2+21)

(iii) n(B - A) = 21

Note: Elements which are in B but not in A (23-2)

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V6Question 6: In the adjoining figure, A and B are two sets of the universal set \xi such that B \subset A \subset \xi , n(A) = 41 , n(B) = 25 and n (]xi) = 50 . Find: (i) n(A')    (ii) n(B')      (iii) n(A - B)


Answers:

(i) n(A') = 50 - 41 = 9    (ii) n(B') = 50 - 25 = 25

(iii) n(A - B) = 41 - 25 = 16

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V7Question 7: If \xi = \{x : x \in N \ and \ x \leq 20 \} , A = \{x : x \ is \ a \ multiple \ of \ 4 \} , B = \{x : x \ is \ a \ multiple \ of 6 \} and C = \{x : x \ is \ a \ factor \ of \ 36 \} . Draw a Venn diagram to show that the relationship between the given sets

(i) A \cap C       (ii) A - B      (iii) A \cap B \cap C

Answer:

The first step is to identify the elements of all the sets. They would be as below:

\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, \ldots , 18, 19, 20 \}

A = \{4, 8, 12, 16, 20 \}

B = \{6, 12, 18 \}

C = \{1, 2, 3, 4, 6, 9, 12, 18, 36 \}

Now it is easy to calculate the following:

(i) A \cap C = \{4, 12 \}      (ii) A - B = \{4, 8, 16, 20 \}      (iii) A \cap B \cap C = \{12 \}

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V8Question 8. In a class of 60 pupils, 28 play hockey, 33 play cricket and 14 play none of these games. Draw the Venn diagram to find:

(i) How many play both games

(ii) How many play hockey only

(iii) How many play cricket only

Answer:

n( \xi ) = 60      n(Hockey) = 28      n(Cricket) = 33

(i) n(\xi) - (n(Hockey) \cup n(Cricket)) = 14

(ii) n(Hockey) - n(Cricket) = 13

(iii) n(Cricket) - n(Hockey) = 18

(iv) n(Hockey) \cap n(Cricket) = 15

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Question 9: In a club, three-tenths of its members play cards only and four-tenths play carom only. If 15 members play none of these games and 90 play both, find using Venn diagram, the total number of members in the club.

V9Answer:

Let the total no. of members n(\xi) = x

n(Cards) = \frac{3x}{10}

n(Carom) = \frac{4x}{10}

n(Cards) \cap n(Carom) = 90

\therefore \frac{3x}{10} + 90 + \frac{4x}{10} + 15 = x

\Rightarrow \frac{7x}{10} + 105 = x

\Rightarrow \frac{3x}{10} = 105

\Rightarrow x = 350

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Question 10: In a colony, two-fifths of the families read the newspaper, “Times of India” and three-fourth of the families read “Hindustan Times”. If 40 families read none of these two newspaper and 100 families read both, use Venn diagram to find the number of families in the colony.V10

Answer:

Now calculate the equation: Let x be the number of families

\Big( \frac{2x}{5} - 100 \Big) + 100 + \Big( \frac{3x}{4} - 100 \Big) + 40 = x

\frac{23x}{20} - x = 60

Hence x = 400

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Question 11: In a class of 50 boys, 35 like horror movies, 30 like war movies, and 5 like neither. Find the number of those who like both.

Answer:V12

Question 12: In a group of persons, each one knows either Hindi or Tamil. If 84 persons know Hindi, 36 know Tamil, and 25 know both, how many people are there in all, in the group?

Answer: Total number of people in the group = 95

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Question 13: In a certain locality of Delhi there are 1000 families. A survey showed that 504 subscribe to “The Hindustan Times” newspaper and 478 subscribe to “The Times of India” newspaper and 106 subscribe to both. Find the number of families that do not subscribe to any of these newspapers.

Answer:

Number of families which do not subscribe to any of these newspapers = 124

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