Q

Q. 11 Let A and B be sets. If A $\cap$ X $=$B $\cap$ X $=$ $\phi$and A $\cup$ X $=$ B $\cup$ X for some set X, show that A $=$B.

D Divya Prakash Singh

Given,  A $\cap$ X $=$B $\cap$ X $=$ $\phi$   and  A $\cup$ X $=$ B $\cup$ X

To prove:   A = B

A = A $\cap$(A$\cup$X)              (A $\cap$ X $=$B $\cap$ X)

= A $\cap$(B$\cup$X)

= (A$\cap$B) $\cup$ (A$\cap$X)

=  (A$\cap$B) $\cup$ $\phi$            (A $\cap$ X $=$ $\phi$)

=  (A$\cap$B)

B = B $\cap$(B$\cup$X)              (A $\cap$ X $=$B $\cap$ X)

= B $\cap$(A$\cup$X)

= (B$\cap$A) $\cup$ (B$\cap$X)

=  (B$\cap$A) $\cup$ $\phi$            (B $\cap$ X $=$ $\phi$)

=  (B$\cap$A)

We know that    (A$\cap$B) =  (B$\cap$A) = A = B

Hence, A = B

Q. 12 Find sets A, B and C such that A $\cap$ B, B $\cap$ C and A $\cap$ C are non-empty sets and A  $\cap$ B $\cap$ C $=$ $\phi$

D Divya Prakash Singh

Given,    A $\cap$ B, B $\cap$ C and A $\cap$ C are non-empty sets

To prove : A  $\cap$ B $\cap$ C $=$ $\phi$

Let A = {1,2}

B = {1,3}

C = {3,2}

Here,   A $\cap$ B = {1}

B $\cap$ C = {3}

A $\cap$ C = {2}

and   A  $\cap$ B $\cap$ C $=$ $\phi$

Q. 13 In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

D Divya Prakash Singh

n ( taking tea) = 150

n (taking coffee) = 225

n ( taking both ) = 100

n(people taking tea or coffee) = n ( taking tea) +  n (taking coffee) -  n ( taking both )

= 150 + 225 - 100

=375 - 100

= 275

Total students = 600

n(students  taking neither tea nor coffee ) = 600 - 275 = 325

Hence,325 students were taking neither tea nor coffee.

Q. 14 In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?

D Divya Prakash Singh

n (hindi ) = 100

n (english ) = 50

n(both) = 25

n(students  in the group ) = n (hindi ) +  n (english ) - n(both)

= 100 + 50 - 25

= 125

Hence,there are 125 students in the group.

Q. 15 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 number of people who read at least one of the newspapers.

(ii) the number of people who read exactly one newspaper.

D Divya Prakash Singh

n(H) = 25

n(T) = 26

n(I) = 26

n(H $\cap$ I) = 9

n( T $\cap$ I ) = 8

n( H $\cap$ T ) = 11

n(H $\cap$ T $\cap$ I ) = 3

the number of people who read at least one of the newspapers = n(H$\cup$T$\cup$I) = n(H) + n(T) + n(I) - n(H $\cap$ I) -  n( T $\cap$ I ) - n( H $\cap$ T ) + n(H $\cap$ T $\cap$ I )

= 25 + 26 + 26 - 9 - 8 - 11 + 3

= 52

Hence, 52 people who read at least one of the newspapers.

(ii) number of people who read exactly one newspaper =

the number of people who read at least one of the newspapers -  n(H $\cap$ I) -  n( T $\cap$ I ) - n( H $\cap$ T ) + 2 n(H $\cap$ T $\cap$ I )

=  52 - 9- 8 -11 + 6

=   30

Hence, 30 number of people who read exactly one newspaper .

Q. 16 In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

D Divya Prakash Singh

n(A) = 21

n(B) = 26

n(C) = 29

n( A $\cap$ B) = 14

n( A $\cap$ C) = 12

n (B $\cap$ C ) = 14

n( A $\cap$ B $\cap$ C) = 8

n(liked product C only) = 29 - 4 -8 - 6 = 11

11 people like only product C.

Q.7 Fill in the blanks to make each of the following a true statement :

(i) A $\cup$ A′ $=$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

(ii) $\phi '$ $\cap$A $=$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

(iii) A $\cap$ A′ $=$$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

(iv) U′ $\cap$ A $=$$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

D Divya Prakash Singh

The following are the answers for the questions

(i) A $\cup$ A′ $=$ U

(ii) $\phi$$\cap$ A  $=$ A

(iii) A $\cap$ A′ $=$ $\phi$

(iv) U′ $\cap$ A $=$ $\phi$

Q. 6 Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from $60\degree$, what is A′?

D Divya Prakash Singh

A' is the set of all triangles whose angle is $60\degree$ in other words A' is set of all equilateral triangles.

Q. 5 Draw appropriate Venn diagram for each of the following :

(i) (A ∪ B)′

(ii) A′ ∩ B′

(iii) (A ∩ B)′

(iv) A′ ∪ B′

D Divya Prakash Singh

(i) (A ∪ B)′

(A ∪ B) is in yellow colour

(A ∪ B)′ is in green colour

ii) A′ ∩ B′ is represented by the green colour in the below figure

iii) (A ∩ B)′ is represented by green colour in the below diagram and white colour represents (A ∩ B)

iv)  A′ ∪ B′

The green colour represents   A′ ∪ B′

Q. 4 If U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}. Verify that

(i) (A $\cup$ B)′ = A′ $\cap$ B′

(ii) (A $\cap$ B)′ = A′ $\cup$ B′

D Divya Prakash Singh

U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}

(i) (A $\cup$ B)′ = A′ $\cap$ B′

L.H.S = (A $\cup$ B)′ = U - (A $\cup$ B) = {1,9}

R.H.S =  A′ $\cap$ B′ = {1,3,5,7,9} $\cap$ {1,4,6,8,9} = {1,9}

L.H.S = R.H.S

Hence,the statement is true.

(ii) (A $\cap$ B)′ = A′ $\cup$ B′

L.H.S = U - (A $\cap$ B) ={1,3,4,5,6,7,8,9}

R.H.S =  A′ $\cup$ B′ = {1,3,5,7,9} $\cup$ {1,4,6,8,9} = {1,3,4,5,6,7,8,9}

L.H.S = R.H.S

Hence,the statement is true.

Q.3 Taking the set of natural numbers as the universal set, write down the complements of the following sets:

(vi) { x : x is a perfect square }

(vii) { x : x is a perfect cube}

(viii) { x : x + 5 $=$8 }

(ix) { x : 2x + 5 $=$9}

(x) { x : x $\geq$ 7 }

(xi) { x : x $\in$N and 2x + 1 $>$ 10 }

D Divya Prakash Singh

Universal set = U = {1,2,3,4,5,6,7,8.............}

(vi) { x : x is a perfect square }' = U - { x : x is a perfect square } =  { x : x $\in$ N and x is not a perfect square }

(vii) { x : x is a perfect cube}' = U - { x : x is a perfect cube} =  { x : x $\in$ N and x is not a perfect cube}

(viii) { x : x + 5 $=$8 }' = U - { x : x + 5 $=$8 } = U - {3} = { x : x$\in$ N and x $\neq$ 3 }

(ix) { x : 2x + 5 $=$9}' = U - { x : 2x + 5 $=$9} = U -{2} =  { x : x$\in$ N and x $\neq$ 2}

(x) { x : x $\geq$ 7 }' = U -  { x : x $\geq$ 7 } =  { x : x$\in$ N and x $<$ 7 }

(xi) { x : x $\in$N and 2x + 1 $>$ 10 }' = U - { x : x $\in$N and x  $>$ 9/2 } =  { x : x$\in$ N and x $\leq$  9/2 }

Q.2 If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :

(i) A = {a, b, c}

(ii) B = {d, e, f, g}

(iii) C = {a, c, e, g}

(iv) D = { f, g, h, a}

D Divya Prakash Singh

U = { a, b, c, d, e, f, g, h}

(i) A = {a, b, c}

A' = U - A = {d,e,f,g,h}

(ii) B = {d, e, f, g}

B' = U - B = {a,b,c,h}

(iii) C = {a, c, e, g}

C' = U - C = {b,d,f,h}

(iv) D = { f, g, h, a}

D' = U - D = {b,c,d,e}

Q. 1 Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and C = { 3, 4, 5, 6 }. Find

(i) A′

(ii) B′

(iii) (A $\cup$ C)′

(iv) (A $\cup$ B)′

(v) (A')'

(vi) (B – C)'

D Divya Prakash Singh

U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and C = { 3, 4, 5, 6 }

(i) A′ = U - A = {5,6,7,8,9}

(ii) B′ = U - B = {1,3,5,7,9}

(iii) A $\cup$ C = {1,2,3,4,5,6}

(A $\cup$ C)′ = U - (A $\cup$ C) = {7,8,9}

(iv) (A $\cup$ B) = {1,2,3,4,6,8}

(A $\cup$ B)′ = U - (A $\cup$ B) = {5,7,9}

(v) (A')' = A = { 1, 2, 3, 4}

(vi) (B – C) = {2,8}

(B – C)' = U - (B – C) = {1,3,4,5,6,7,9}

Q. 4 If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

(VII) B $\cup$ C $\cup$ D

S safeer

Here,B = {3, 4, 5, 6},

C = {5, 6, 7, 8 } and

D = { 7, 8, 9, 10 }

The union can be written as

B $\cup$ C $\cup$ D = {3,4,5,6,7,8,9,10}

Q.4 If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

(VI) A $\cup$ B $\cup$ D

S safeer

Here,

A = {1, 2, 3, 4},

B = {3, 4, 5, 6}

D = { 7, 8, 9, 10 }

The union can be written as

$\cup$ B $\cup$ D = {1,2,3,4,5,6,7,8,9,10}

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