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) $A \cup B$

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Question 2. From the Adjoining figure, find:

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

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

(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:

(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)    $(A \cup B \cup C)'$

(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:

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

(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:

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

(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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Question 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) $n(A - B)$

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

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

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Question 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) $A \cap B \cap C$

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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Question 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

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

(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.

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

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

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

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

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

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

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

$\Rightarrow x = 350$

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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.

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

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

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

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.

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 $= 95$
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.
Number of families which do not subscribe to any of these newspapers $= 124$