Class 8: Venn Diagrams

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:

$\\$

$\displaystyle \textbf{Question 2: }\text{From the adjoining figure, find:}$
$\displaystyle \text{(i) }A\cap B\qquad\text{(ii) }A\cup B\qquad\text{(iii) }A'$
$\displaystyle \text{(iv) }B'\qquad\text{(v) }(A\cup B)'\qquad\text{(vi) }(A\cap B)'$

$\displaystyle \text{Answer:}$

$\displaystyle \text{(i) }A\cap B=\{10,13\}$
$\displaystyle \text{Note: Take the elements which are common to both }A\text{ and }B.$

$\displaystyle \text{(ii) }A\cup B=\{10,11,12,13,16,19\}$
$\displaystyle \text{Note: Take all the elements which are in }A\text{ or }B\text{ or in both.}$

$\displaystyle \text{(iii) }A'=\{11,12,14,15,17,18\}$
$\displaystyle \text{Note: Take all the elements of the universal set which are not in }A.$

$\displaystyle \text{(iv) }B'=\{14,15,16,17,18,19\}$
$\displaystyle \text{Note: Take all the elements of the universal set which are not in }B.$

$\displaystyle \text{(v) }(A\cup B)'=\{14,15,17,18\}$
$\displaystyle \text{Note: Take all the elements of the universal set which are not in }A\cup B.$

$\displaystyle \text{(vi) }(A\cap B)'=\{11,12,14,15,16,17,18,19\}$
$\displaystyle \text{Note: Take all the elements of the universal set which are not in }A\cap B.$

$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{Use the adjoining figure to find the following:}$
$\displaystyle \text{(i) }A\cap B\qquad\text{(ii) }A\cap C\qquad\text{(iii) }B\cap C$
$\displaystyle \text{(iv) }A\cap B\cap C\qquad\text{(v) }C'\qquad\text{(vi) }A\cup B$
$\displaystyle \text{(vii) }A-B\qquad\text{(viii) }B-C\qquad\text{(ix) }C-A$
$\displaystyle \text{(x) }(B\cup C)'\qquad\text{(xi) }(A\cup C)'\qquad\text{(xii) }(A\cup B\cup C)'$

$\displaystyle \text{Answer:}$

$\displaystyle \text{(i) }A\cap B=\{2,4\}$

$\displaystyle \text{(ii) }A\cap C=\{4,5\}$

$\displaystyle \text{(iii) }B\cap C=\{4,6\}$

$\displaystyle \text{(iv) }A\cap B\cap C=\{4\}$

$\displaystyle \text{(v) }C'=\{2,3,9,10,11\}$

$\displaystyle \text{(vi) }A\cup B=\{2,3,4,5,6,10\}$

$\displaystyle \text{(vii) }A-B=\{5,10\}$

$\displaystyle \text{(viii) }B-C=\{2,3\}$

$\displaystyle \text{(ix) }C-A=\{6,7,8\}$

$\displaystyle \text{(x) }(B\cup C)'=\{9,10,11\}$

$\displaystyle \text{(xi) }(A\cup C)'=\{3,9,11\}$

$\displaystyle \text{(xii) }(A\cup B\cup C)'=\{9,11\}$

$\displaystyle \\$

$\displaystyle \textbf{Question 4: }\text{Use the adjacent Venn diagram to find:}$
$\displaystyle \text{(i) }A\cap B\qquad\text{(ii) }A\cup B\qquad\text{(iii) }B-A$
$\displaystyle \text{(iv) }A-B\qquad\text{(v) }A\qquad\text{(vi) }A'$

$\displaystyle \text{Answer:}$

$\displaystyle \text{(i) }A\cap B=\{2,4\}$

$\displaystyle \text{(ii) }A\cup B=\{2,4,5,7,9\}$

$\displaystyle \text{(iii) }B-A=\phi$
$\displaystyle \text{Note: There are no elements in }B\text{ which are not in }A.\text{ Hence, }B-A=\phi.$

$\displaystyle \text{(iv) }A-B=\{5,7,9\}$

$\displaystyle \text{(v) }A=\{2,4,5,7,9\}$

$\displaystyle \text{(vi) }A'=\{1,3,6,8,10\}$

$\displaystyle \\$

$\displaystyle \textbf{Question 5: }\text{Let }A\text{ and }B\text{ be sets such that }n(A)=17,\ n(A\cup B)=38$
$\displaystyle \text{and }n(A\cap B)=2.\text{ Draw a Venn diagram and find:}$
$\displaystyle \text{(i) }n(A-B)\qquad\text{(ii) }n(B)\qquad\text{(iii) }n(B-A)$

$\displaystyle \text{Answer:}$

$\displaystyle \text{Given, }n(A)=17,\quad n(A\cup B)=38,\quad n(A\cap B)=2.$

$\displaystyle \text{(i) }n(A-B)=n(A)-n(A\cap B)=17-2=15.$
$\displaystyle \text{Note: These are the elements which are in }A\text{ but not in }B.$

$\displaystyle \text{(ii) }n(B)=23$
$\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B).$
$\displaystyle 38=17+n(B)-2.$
$\displaystyle \therefore n(B)=23.$
$\displaystyle \text{Note: The Venn diagram has }21\text{ elements only in }B\text{ and }2\text{ common elements.}$
$\displaystyle \therefore n(B)=21+2=23.$

$\displaystyle \text{(iii) }n(B-A)=n(B)-n(A\cap B)=23-2=21.$
$\displaystyle \text{Note: These are the elements which are in }B\text{ but not in }A.$

$\displaystyle \\$

$\displaystyle \textbf{Question 6: }\text{In the adjoining figure, }A\text{ and }B\text{ are two subsets of the universal}$
$\displaystyle \text{set }\xi\text{ such that }B\subset A\subset\xi,\ n(A)=41,\ n(B)=25\text{ and }n(\xi)=50.$
$\displaystyle \text{Find: (i) }n(A')\qquad\text{(ii) }n(B')\qquad\text{(iii) }n(A-B)$

$\displaystyle \text{Answer:}$

$\displaystyle \text{(i) }n(A')=n(\xi)-n(A)=50-41=9.$

$\displaystyle \text{(ii) }n(B')=n(\xi)-n(B)=50-25=25.$

$\displaystyle \text{(iii) }n(A-B)=n(A)-n(B)=41-25=16.$

$\displaystyle \\$

$\displaystyle \textbf{Question 7: }\text{If }\xi=\{x:x\in N\text{ and }x\leq20\},$
$\displaystyle A=\{x:x\text{ is a multiple of }4\},\quad B=\{x:x\text{ is a multiple of }6\}\text{ and}$
$\displaystyle C=\{x:x\text{ is a factor of }36\}.\text{ Draw a Venn diagram to show the relationship}$
$\displaystyle \text{between the given sets and find:}$
$\displaystyle \text{(i) }A\cap C\qquad\text{(ii) }A-B\qquad\text{(iii) }A\cap B\cap C$

$\displaystyle \text{Answer:}$

$\displaystyle \text{The first step is to identify the elements of all the sets.}$

$\displaystyle \xi=\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}.$

$\displaystyle A=\{4,8,12,16,20\}.$

$\displaystyle B=\{6,12,18\}.$

$\displaystyle C=\{1,2,3,4,6,9,12,18\}.$

$\displaystyle \text{Now, we can calculate the following:}$

$\displaystyle \text{(i) }A\cap C=\{4,12\}.$

$\displaystyle \text{(ii) }A-B=\{4,8,16,20\}.$

$\displaystyle \text{(iii) }A\cap B\cap C=\{12\}.$

$\displaystyle \\$

$\displaystyle \textbf{Question 8: }\text{In a class of }60\text{ pupils, }28\text{ play hockey, }33\text{ play cricket}$
$\displaystyle \text{and }14\text{ play neither of these games. Draw a Venn diagram to find:}$
$\displaystyle \text{(i) How many play both games}$
$\displaystyle \text{(ii) How many play hockey only}$
$\displaystyle \text{(iii) How many play cricket only}$

$\displaystyle \text{Answer:}$

$\displaystyle \text{Let }H\text{ be the set of pupils who play hockey and }C\text{ be the set of pupils who play cricket.}$
$\displaystyle n(\xi)=60,\qquad n(H)=28,\qquad n(C)=33.$

$\displaystyle \text{Number of pupils who play neither game}=14.$
$\displaystyle \therefore n(H\cup C)=60-14=46.$

$\displaystyle \text{Using }n(H\cup C)=n(H)+n(C)-n(H\cap C),$
$\displaystyle 46=28+33-n(H\cap C).$
$\displaystyle \therefore n(H\cap C)=28+33-46=15.$

$\displaystyle \text{(i) Number of pupils who play both games}=15.$

$\displaystyle \text{(ii) Number of pupils who play hockey only}=n(H)-n(H\cap C)$
$\displaystyle =28-15=13.$

$\displaystyle \text{(iii) Number of pupils who play cricket only}=n(C)-n(H\cap C)$
$\displaystyle =33-15=18.$

$\displaystyle \\$

$\displaystyle \textbf{Question 9: }\text{In a club, three-tenths of its members play cards only and four-tenths}$
$\displaystyle \text{play carom only. If }15\text{ members play neither of these games and }90\text{ play both,}$
$\displaystyle \text{find, using a Venn diagram, the total number of members in the club.}$

$\displaystyle \text{Answer:}$

$\displaystyle \text{Let the total number of members in the club be }n(\xi)=x.$

$\displaystyle \text{Number of members who play cards only}=\frac{3x}{10}.$

$\displaystyle \text{Number of members who play carom only}=\frac{4x}{10}.$

$\displaystyle \text{Number of members who play both games}=90.$

$\displaystyle \text{Number of members who play neither game}=15.$

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

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

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

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

$\displaystyle \Rightarrow x=\frac{105\times10}{3}=350.$

$\displaystyle \therefore \text{The total number of members in the club is }350.$

$\displaystyle \\$

$\displaystyle \textbf{Question 10: }\text{In a colony, two-fifths of the families read the newspaper}$
$\displaystyle \text{''Times of Indi'' and three-fourths of the families read''Hindustan Times''}$
$\displaystyle \text{If }40\text{ families read neither newspaper and }100\text{ families read both, use a Venn}$
$\displaystyle \text{diagram to find the number of families in the colony.}$

$\displaystyle \text{Answer:}$ V10

$\displaystyle \text{Let the total number of families in the colony be }x.$

$\displaystyle \text{Number of families reading Times of India}=\frac{2x}{5}.$

$\displaystyle \text{Number of families reading Hindustan Times}=\frac{3x}{4}.$

$\displaystyle \text{Since }100\text{ families read both newspapers,}$

$\displaystyle \text{Times of India only}=\frac{2x}{5}-100.$

$\displaystyle \text{Hindustan Times only}=\frac{3x}{4}-100.$

$\displaystyle \text{Number of families reading neither newspaper}=40.$

$\displaystyle \therefore \left(\frac{2x}{5}-100\right)+100+\left(\frac{3x}{4}-100\right)+40=x.$

$\displaystyle \Rightarrow \frac{2x}{5}+\frac{3x}{4}-60=x.$

$\displaystyle \Rightarrow \frac{8x+15x}{20}-60=x.$

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

$\displaystyle \Rightarrow \frac{3x}{20}=60.$

$\displaystyle \Rightarrow x=400.$

$\displaystyle \therefore \text{The total number of families in the colony is }400.$

$\displaystyle \\$

$\displaystyle \textbf{Question 11: }\text{In a class of }50\text{ boys, }35\text{ like horror movies, }30\text{ like war}$
$\displaystyle \text{movies, and }5\text{ like neither. Find the number of those who like both.}$

$\displaystyle \text{Answer:}$ V12

$\displaystyle \text{Let }H\text{ be the set of boys who like horror movies and }W\text{ be the set of boys who like war movies.}$

$\displaystyle n(\xi)=50,\qquad n(H)=35,\qquad n(W)=30.$

$\displaystyle \text{Since }5\text{ boys like neither, }n(H\cup W)=50-5=45.$

$\displaystyle \text{Using }n(H\cup W)=n(H)+n(W)-n(H\cap W),$

$\displaystyle 45=35+30-n(H\cap W).$

$\displaystyle \Rightarrow n(H\cap W)=35+30-45=20.$

$\displaystyle \therefore \text{The number of boys who like both horror and war movies is }20.$

$\displaystyle \\$

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

$\displaystyle \text{Answer:}$ V13

$\displaystyle \text{Let }H\text{ be the set of persons who know Hindi and }T\text{ be the set of persons who know Tamil.}$

$\displaystyle n(H)=84,\qquad n(T)=36,\qquad n(H\cap T)=25.$

$\displaystyle \text{Using }n(H\cup T)=n(H)+n(T)-n(H\cap T),$

$\displaystyle n(H\cup T)=84+36-25=95.$

$\displaystyle \therefore \text{The total number of people in the group is }95.$

$\displaystyle \\$

$\displaystyle \textbf{Question 13: }\text{In a certain locality of Delhi, there are }1000\text{ families. A survey}$
$\displaystyle \text{showed that }504\text{ subscribe to ''The Hindustan Times'' newspaper and }478\text{ subscribe}$
$\displaystyle \text{to ''The Times of India'' newspaper, while }106\text{ subscribe to both. Find the number}$
$\displaystyle \text{of families that do not subscribe to either of these newspapers.}$

$\displaystyle \text{Answer:}$

$\displaystyle \text{Let }H\text{ be the set of families subscribing to The Hindustan Times and }T\text{ be the set}$
$\displaystyle \text{of families subscribing to The Times of India.}$ V14

$\displaystyle n(\xi)=1000,\qquad n(H)=504,\qquad n(T)=478,\qquad n(H\cap T)=106.$

$\displaystyle \text{Using }n(H\cup T)=n(H)+n(T)-n(H\cap T),$

$\displaystyle n(H\cup T)=504+478-106=876.$

$\displaystyle \text{Number of families subscribing to neither newspaper}$
$\displaystyle =n(\xi)-n(H\cup T)=1000-876=124.$

$\displaystyle \therefore \text{The number of families that do not subscribe to either newspaper is }124.$

$\displaystyle \\$

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4 thoughts on “Class 8: Venn Diagrams – Exercise 2”

tiasaa sarkar says:

February 6, 2018 at 3:02 pm

thank you for your help

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ICSE Student-Assistant says:

September 22, 2018 at 12:05 pm

Could you point me the exercise and the problem number…. we will fix it. thanks for pointing out errors so that we can fix it.

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Class 8: Venn Diagrams – Exercise 2

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