Question 1: If A and B are two sets such that n(A \cup B) =50, n(A) =28 and n(B) = 32 , find n(A \cap B) .

Answer:

2019-04-30_8-54-04Given A ( \cup A ) = 50, \ n( A ) = 28, \ n( B ) = 32

We know: n ( A \cup B ) = n(A) + n ( B ) - n ( A \cap B)

\Rightarrow n ( A \cap B) = n( A) + n ( B )  - n ( A \cup B )

\Rightarrow n ( A \cap B ) = 28 + 32 - 50

\Rightarrow n ( A \cap B ) = 10

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Question 2: If P and Q are two sets such that P has 40 elements, P \cup Q has 50 elements and P \cap Q has 10 elements, how many elements does Q have?

Answer:

2019-04-30_8-55-06Given: n(P ) = 40, \ n ( P \cup Q) = 60 , \ n ( P \cap Q) = 10

We know: n ( P \cup Q ) = n(P) + n ( Q ) - n ( P \cap Q)

\Rightarrow 60 = 40 + n(Q) -10

\Rightarrow n(Q) = 70 - 40 = 30

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Question 3: In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4  teach physics and mathematics. How many teach physics?

Answer:

2019-04-30_8-55-19Given: n (M \cup P) = 20, \ n(M) = 12, \ n(M \cap P) = 4

We know: n ( M \cup P ) = n(M) + n ( P ) - n ( M \cap P)

\Rightarrow 20 = 12 + n(P) - 4

\Rightarrow n(P) = 12

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Question 4: In a group of 70 people, 37 like coffee, 52 like tea and each person likes at east one of the two drinks. How many like both coffee and tea?

Answer:

2019-04-30_9-07-18Given: n ( T \cup C ) = 70, \ n (T) = 52, \ n (C ) = 37

We know: n ( T \cup C ) = n(T) + n ( C ) - n ( T \cap C)

\Rightarrow 70 = 52 + 37 - n (T \cap C)

\Rightarrow T ( T \cap C) = 19

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Question 5: Let A and B be two sets such that: n(A) =20, n(A \cup B) = 42 and n(A \cap B) = 4 . Find : (i) n(A)    (ii) n(A-B)    (iii) n(B-A)

Answer:

2019-04-30_9-07-33Given: n ( A ) = 20, \ n (A \cup B) = 42, \ n(A \cap B) = 4

i) n ( A \cup B ) = n ( A ) + n ( B )  - n ( A \cap B)

\Rightarrow 42 = 20 + n ( B ) - 4

\Rightarrow n ( B ) = 26

ii) n ( A - B ) = n ( A ) - n ( A \cap B )

\Rightarrow n ( A - B )= 20 - 4 = 16

iii) n ( B - A ) = n ( B ) - n ( B \cap A )

\Rightarrow n ( B - A ) = 26 - 4 = 22

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Question 6: A survey shows that 76\% of the Indians like oranges, whereas 62\% like bananas. What percentage of the Indians like both oranges and bananas?

Answer:

2019-05-01_8-16-14.pngLet the population be 100

Therefore n ( O \cup B ) = 100

Also n ( O ) = 76, \ n ( B ) = 62

We know: n ( O \cup B ) = n(O) + n ( B ) - n ( O \cap B )

\Rightarrow 100 = 76 + 62 - n ( O \cap B )

\Rightarrow n ( O \cap B ) = 38

Therefore 38\% of population likes both Banana and Oranges.

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Question 7: In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: (i) how many can speak both Hindi and English (ii) how many can speak Hindi only (iii) how many can speak English only

Answer:

2019-05-01_8-30-28Given: n ( H ) = 750, \ n ( E ) = 460, \ n ( H \cup E ) = 950

i) We know: n ( H \cup E ) = n(H) + n ( E ) - n ( H \cap E )

\Rightarrow 950 = 750 + 460 - n ( H \cap E )

\Rightarrow n ( H \cap E ) = 260

Therefore 260 people can speak both Hindi and English

ii) n( H ) = n ( H - E ) + n ( H \cap E)

\Rightarrow 750 = n ( H - E ) + 260

\Rightarrow n ( H - E ) = 750 -260 = 490

Therefore 490 people can speak Hindi only

iii) n ( E ) = n ( E - H ) + n ( E \cap H)

\Rightarrow 460 = n ( E - H ) + 260

\Rightarrow n ( E - H ) = 200

Therefore 200 people speak English only.

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Question 8: In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find: (i) how many drink tea and coffee both (ii) how many drink coffee but not tea.

Answer:

2019-05-01_8-31-07Given: n(T) = 30, \ n(T-C) = 14, \ n(T \cup C) = 50

i) n(T) = n(T-C) + n ( T \cap C)

\Rightarrow 30 = 14 + n ( T \cap C)

\Rightarrow n(T \cap C) = 16

Therefore 16 people drink both Tea and Coffee.

ii) n( T \cup C) = n(T) + n(C) - n(T \cap C)

\Rightarrow 50 = 30 + n(C) - 16

\Rightarrow n(C) = 36

Now we need to find n(C-T)

n(C) = n(C - T) + n ( C \cap T)

\Rightarrow 36 = n ( C - T) + 16

\Rightarrow n(C-T) = 20

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Question 9: In a survey of 60 people, it was found that 25 people read newspaper H, \ 26 read newspaper T, \ 26 read newspaper I, \ 9 read both H and I, \ 11 read both H and T, \ 8 read both T and I, \ 3 read all three newspapers. Find: (i) the numbers of people who read at least one of the newspapers. (ii) the number of people who read exactly one newspaper.

Answer:

Given: n(H) = 25,  \ n(T) = 26, \ n(I) = 26 n( H \cap I) = 9, \ n( H \cap T) = 11, \ n( T \cap I) = 8, \ n ( H \cap T \cap I) = 3

i) n( H \cup T \cup I ) = n(H) + n(T) + n(I) - n( H \cap T) - n( T \cap I) - n( I \cap H) + n ( H \cap T \cap I) 

2019-05-01_8-49-04= 25 + 26 + 26 - 9 - 11- 8 + 3

= 52

Note: There are 8 people who do no read any news paper. We should not assume that all read newspaper.

ii)

n( H \ or \ T \ or \ I) = n(H) + n(T) + n(I) - 2n(H \cap T) - 2n(H \cap I) - 2n(T \cap I)+ 3n(H \cap T \cap I)

= 25 + 26 + 26 - 22- 16- 18 + 9 

= 30

You can also do it by looking at the venn diagram. Venn diagram is more intutitive.

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Question 10: Of the members of three athletic teams in a certain school, 21 are in the basketball team, 25 in hockey team and 29 in the football team. 14 play hockey and basket ball, 15 play hockey and football, 12 play football and basketball and 8 play all the three games. How many members are there in all?

Answer:2019-05-01_8-55-34

Given: n(B) = 21, \ n(H) = 26, \ n(F) = 29 n(H \cap B) = 14

n(H \cap F) = 15 n(H \cap B \cap F) = 8

We know: n( H \cup B \cup F ) = n(H) + n(B) + n(F) - n( H \cap B) - n( B \cap F) - n( F \cap H) + n ( H \cap B \cap F) 

\Rightarrow n ( H \cap B \cap F) = 26 + 21 + 29 -14 - 15 - 12 + 8 = 43

Therefore total number of members in all the teams are 43

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Question 11: In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali ? How many can speak both Hindi and Bengali?

Answer:

Given: n ( H \cup B) = 1000, \ n(H) = 750, \ n(B) = 400

We know: n ( H \cup B ) = n(H) + n (B ) - n ( H \cap B )

\Rightarrow 1000 = 750+400 - n ( H \cap B) 2019-05-01_9-04-05.png

\Rightarrow n ( H \cap B ) = 150

Therefore the number of people who can speak both Hindi and Bengali is 150 .

Number of people who can speak only Hindi

n ( H - B ) = n ( H ) - n ( H \cap B)  = 750 - 150 = 600

No of people who only speak Bengali

n ( B - H ) = n(B) - n ( B \cap H )  = 400 - 150 = 250

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Question 12: A survey of 500 television viewers produced the following in-formation; 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, 50 do not watch any of the three games. How many watch all the three games? How many watch exactly one of the three games?

Answer:

Given: n(F) = 285, \ n ( H ) = 195 , \ n(B) = 115 n ( F \cap B) = 45 , \ n( F \cap H) = 70, \ n( H \cap B) = 50

n ( F' \cap H' \cap B') = 50 2019-05-01_9-22-25.png

\Rightarrow n ( F \cup H \cup B) ' = 50

\Rightarrow n ( U ) - n ( F \cup H \cup B) = 50

\Rightarrow 500 - n ( F \cup H \cup B) = 50

\Rightarrow n ( F \cup H \cup B) = 450

Now, the number of students who watch all the games

n ( F \cap H \cap B) = n ( F \cup H \cup B) - n(F) - n(H) - n(B) + n ( F \cap B) + n ( F \cap H) + n (H \cap B)

= 450 - 285 - 195 - 115 + 45 + 70 + 50  = 20

No of students who watch exactly one game

= n ( F) + n( H ) + n(B) - 2n( F \cap B) - 2n ( H \cap B ) - 2n ( H \cap F) + 3n ( H \cap B \cap F)

= 285 + 195 + 115 - 90 - 140 - 100 + 60 = 325

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Question 13: In a survey of 100 persons it was found that 28 read magazine A, \ 30 read magazine B, \ 42 read magazine C, \ S  read magazines A and B, \ 10 read magazines A and C, \ S read magazines B and C and 3 read all the three magazines. Find: (i) How many read none of three magazines? (ii) How many read magazine C only?

Answer:

Given: n ( A ) = 28, \ n ( B ) = 30 , \ n(C) = 42 n ( A \cap B)= 8  , \ n( A \cap  C) = 10 , \ n ( B \cap  C) = 5 \ \      n ( A \cap  B \cap  C) = 3

i) Number of people who read none of the three magazines

n ( A \cup B \cup C) = n ( U ) - n ( A \cup B \cup C)

= n ( U ) - \{ n(A) + n(B) + n(C) - n ( A \cap  B) - n ( B \cap  C) - n ( C \cap  A) + n ( A \cap  B \cap  C) \}

= 100 - \{ 28 + 30 + 42 - 8 - 10 - 5 + 3 \}

= 100 - 80 = 20

ii) Number of students who read C only = n ( C \cap  A' \cap B')

= n \{ C \cap  ( A \cup B)' \} 2019-05-01_9-34-12

= n(C) - n \{ C \cap (A \cup B ) \}

= n(C) - n \{ (C \cap  A) \cup ( C \cap  B ) \}

= n(C) - n \{ (C \cap  A) + (  C \cap B) - ( A \cap B \cap  C) \}

= 42 - ( 10 + 5 - 3) = 42 - 12 = 30

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Question 14: In a survey of 100 students, the number of students studying the various languages were found to be: English only 18 , English but not Hindi 23 , English and Sanskrit 8 , English 26 , Sanskrit 48 , Sanskrit and Hindi 8 , no language 24 . Find:
(i) How many students were studying Hindi?
(ii) How many students were studying English and Hindi?

Answer:

Given:  n ( E) = 26, \ n(S) = 48 , \ n(E \cap S) = 8

n ( E \cap H' \cap S') = 18, \ n(E \cap H') = 23, \ n( E' \cap H' \cap S') = 24

ii) n ( E \cap H') = n(E) - n ( E \cap H )

\Rightarrow 23 = 26 - n ( E \cap H) 2019-05-01_9-56-41.png

\Rightarrow n ( E \cap H ) = 3

i) n( E \cap H' \cap S') = n( E) - n \{ E \cap ( H \cup S') \}

\Rightarrow 18 = 26 - n \{ ( E \cap H ) \cup ( E \cap S) \}

\Rightarrow 18 = 26 - \{ 3 + 8 - n (E \cap H \cap S ) \}

\Rightarrow n ( E \cap H \cap S) = 3

n ( E' \cap H' \cap S') = n(U) - n ( E \cup H \cup S)

24 = 100 - n ( E \cup H \cup S)

Therefore number of students studying Hindi

= n( E \cup H \cup S) - n ( E) - n ( S) + n ( E \cap H ) + n ( E \cap S) + n ( S \cap H) - n ( E \cap H \cap S)

= 76 -24 -48 + 3 + 8 + 8 - 3 = 18

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Question 15: In a survey it was found that 21 persons liked product P_1, \ 26 liked product P_2 and 29 liked product P_3 . If 14  persons liked products P_1 and P_2; \ 12 persons liked product P_3 and P_1; \ 14 persons liked products P_2 and P_3  and 8 liked all the three products. Find how many liked product P_3 only.

Answer:

Given: n(P_1) = 21, \ n ( P_2) = 26, \ n ( P_3) = 29

n ( P_1 \cap P_2) = 14, \ n ( P_3 \cap P_1 ) = 12, \ n ( P_2 \cap P_3) = 14

n ( P_1 \cap P_2 \cap P_3) = 8 2019-05-01_10-20-09

Number of people liking Product P_3 only

= n ( P_3 \cap {P_1}' \cap {P_2}' )

= n \{ P_3 \cap ( P_1 \cup P_2)' \}

= n ( P_3) - n \{ P_3 \cap ( P_1 \cup P_2) \}

= n ( P_3) - n \{ ( P_3 \cap P_1) \cup ( P_3 \cap P_2 \}

= n ( P_3) - \{ n (P_3 \cap P_1) + n (P_3 \cap P_2) - n ( P_1 \cap P_2 \cap P_3 ) \}

= 29 - ( 12 + 14 - 8 ) = 11

Therefore number of people liking P_3 is 11 

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Question 16: A market research group conducted survey on 1000 persons and reported that 720 persons liked product A  and 450 persons liked product B . What is the least number of persons that must have liked both products?

Answer:

Given: n ( A) = 720, \ n(B) = 450, \ n ( A \cup B) = 1000 2019-05-01_10-20-46

We know: n ( A \cup B) = n (A) + n(B) - n(A \cap B)

\Rightarrow 1000 = 750 + 450 - n ( A \cap B)

\Rightarrow n ( A \cap B) = 170

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