NCERT Solutions for Miscellaneous Exercise Chapter 1 Class 11 - Sets

Access Sets Class 11 Chapter 1-Miscellaneous Exercise

Question:1 Decide, among the following sets, which sets are subsets of one and another:

A = { x : x R and x satisfy – 8x + 12 = 0 }, B = { 2, 4, 6 },

C = { 2, 4, 6, 8, . . . }, D = { 6 }.

Answer:

Solution of this equation are – 8x + 12 = 0

( x - 2 ) ( x - 6 ) = 0

X = 2,6

A = { 2,6 }

B = { 2, 4, 6 }

C = { 2, 4, 6, 8, . . . }

D = { 6 }

From the sets given above, we can conclude that A B, A C, D A, D B, D C, B C

Hence, we can say that D A B C

Question:2(i) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If x A and A B , then x B

Answer:

The given statment is false,

example: Let A = { 2,4 }

B = { 1,{2,4},5}

x be 2.

then, 2 { 2,4 } = x A and { 2,4 } { 1,{2,4},5} = A B

But 2 { 1,{2,4},5} i.e. x B

Question:2(ii) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A B and B C , then A C

Answer:

The given statement is false,

Let , A = {1}

B = { 1,2,3}

C = {0,{1,2,3},4}

Here, {1} { 1,2,3} = A B and { 1,2,3} {0,{1,2,3},4} = B C

But, {1} {0,{1,2,3},4} = A C

Question:2(iii) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A B and B C , then A C

Answer:

Let A ⊂ B and B ⊂ C

There be a element x such that

Let, x A

x B ( Because A B )

x C ( Because B C )

Hence,the statement is true that A C

Question:2(iv) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A B and B C , then A C

Answer:

The given statement is false

Let , A = {1,2}

B = {3,4,5 }

C = { 1,2,6,7,8}

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

But , {1,2} { 1,2,6,7,8} = A C

Question:2(v) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If x A and A B , then x B

Answer:

The given statement is false,

Let x be 2

A = { 1,2,3}

B = { 4,5,6,7}

Here, 2 { 1,2,3} = x A and { 1,2,3} { 4,5,6,7} = A B

But, 2 { 4,5,6,7} implies x B

Question:2(vi) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A B and x B , then x A

Answer:

The given statement is true,

Let, A B and x B

Suppose, x A

Then, x B , which is contradiction to x B

Hence, x A.

Question:3 Let A, B, and C be the sets such that A B = A C and A B = A C. Show that B = C.

Answer:

Let A, B, and C be the sets such that A B = A C and A B = A C

To prove : B = C.

A B = A + B - A B = A C = A + C - A C

A + B - A B = A + C - A C

B - A B = C - A C ( since A B = A C )

B = C

Hence proved that B= C.

Question:4 Show that the following four conditions are equivalent :

(i) A B(ii) A – B = (iii) A B = B (iv) A B = A

Answer:

First, we need to show A B A – B =

Let A B

To prove : A – B =

Suppose A – B

this means, x A and x B , which is not possible as A B .

SO, A – B = .

Hence, A B A – B = .

Now, let A – B =

To prove : A B

Suppose, x A

A – B = so x B

Since, x A and x B and A – B = so A B

Hence, A B A – B = .

Let A B

To prove : A B = B

We can say B A B

Suppose, x A B

means x A or x B

If x A

since A B so x B

Hence, A B = B

and If x B then also A B = B.

Now, let A B = B
To prove : A B

Suppose : x A

A A B so x A B

A B = B so x B

Hence,A B

ALSO, A B A B = B

NOW, we need to show A B A B = A

Let A B

To prove : A B = A

Suppose : x A

We know A B A

x A B Also ,A A B

Hence, A B = A

Let A B = A

To prove : A B

Suppose : x A

x A B ( replacing A by A B )

x A and x B

A B

A B A B = A

Question:5 Show that if A B, then C – B C – A.

Answer:

Given , A B

To prove : C – B C – A

Let, x C - B means x C but xB

A B so x C but xA i.e. x C - A

Hence, C – B C – A

Question:6 Assume that P ( A ) = P ( B ). Show that A = B

Answer:

Given, P ( A ) = P ( B )

To prove : A = B

Let, x A

A P ( A ) = P ( B )

For some C P ( B ) , x C

Here, C B

Therefore, x B and A B

Similarly we can say B A.

Hence, A = B

Question:7 Is it true that for any sets A and B, P ( A ) ∪ P ( B ) = P ( A ∪ B )? Justify your answer.

Answer:

No, it is false.

To prove : P ( A ) ∪ P ( B ) P ( A ∪ B )

Let, A = {1,3} and B = {3,4}

A B = {1,3,4}

P(A) = { {},{1},{3},{1,3}}

P(B) = { {},{3},{4},{3,4}}

L.H.S = P(A) P(B) = { {},{1},{3},{1,3},{3,4},{4}}

R.H.S.= P(A B) = { {},{1},{3},{1,3},{3,4},{4},{1,4},{1,3,4}}

Hence, L.H.S. R.H.S

Question:8 Show that for any sets A and B,

A = ( A B ) ( A – B ) and A ( B – A ) = ( A B )

Answer:

A = ( A B ) ( A – B )

L.H.S = A = Red coloured area

R.H.S = ( A B ) ( A – B )

( A B ) = green coloured

( A – B ) = yellow coloured

( A B ) ( A – B ) = coloured part

Hence, L.H.S = R.H.S = Coloured part

A ( B – A ) = ( A B )

A = sky blue coloured

( B – A )=pink coloured

L.H.S = A ( B – A ) = sky blue coloured + pink coloured

R.H.S = ( A B ) = brown coloured part

L.H.S = R.H.S = Coloured part

Question:9(i) Using properties of sets, show that

A ( A B ) = A

Answer:

(i) A ( A B ) = A

We know that A A

and A B A

A ( A B ) A

and also , A A ( A B )

Hence, A ( A B ) = A

Question:9(ii) Using properties of sets, show that

A ( A B ) = A

Answer:

This can be solved as follows

(ii) A ( A B ) = A

A ( A B ) = (A A) ( A B )

A ( A B ) = A ( A B ) { A ( A B ) = A proved in 9(i)}

A ( A B ) = A

Question:10 Show that A B = A C need not imply B = C.

Answer:

Let, A = {0,1,2}

B = {1,2,3}

C = {1,2,3,4,5}

Given, A B = A C

L.H.S : A B = {1,2}

R.H.S : A C = {1,2}

and here {1,2,3} {1,2,3,4,5} = B C.

Hence, A B = A C need not imply B = C.

Question:11 Let A and B be sets. If A X B X and A X B X for some set X, show that A B.

Answer:

Given, A X B X and A X B X

To prove: A = B

A = A (AX) (A X B X)

= A (BX)

= (AB) (AX)

= (AB) (A X )

= (AB)

B = B (BX) (A X B X)

= B (AX)

= (BA) (BX)

= (BA) (B X )

= (BA)

We know that (AB) = (BA) = A = B

Hence, A = B

Question:12 Find sets A, B and C such that A B, B C and A C are non-empty sets and A B C

Answer:

Given, A B, B C and A C are non-empty sets

To prove : A B C

Let A = {1,2}

B = {1,3}

C = {3,2}

Here, A B = {1}

B C = {3}

A C = {2}

and A B C

Question: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?

Answer:

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.

Question: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?

Answer:

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.

Question: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.

Answer:

n(H) = 25

n(T) = 26

n(I) = 26

n(H I) = 9

n( T I ) = 8

n( H T ) = 11

n(H T I ) = 3

the number of people who read at least one of the newspapers = n(HTI) = n(H) + n(T) + n(I) - n(H I) - n( T I ) - n( H T ) + n(H T 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 I) - n( T I ) - n( H T ) + 2 n(H T I )

= 52 - 9- 8 -11 + 6

= 30

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

Question: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.

Answer:

n(A) = 21

n(B) = 26

n(C) = 29

n( A B) = 14

n( A C) = 12

n (B C ) = 14

n( A B C) = 8

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

11 people like only product C.