NCERT Solutions for Miscellaneous Exercise Chapter 1 Class 11 - Sets

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

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

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

Answer:

Solution of this equation are $x^2$ – 8x + 12 = 0

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

X = 2,6

$\therefore$ A = { 2,6 }

B = { 2, 4, 6 }

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

D = { 6 }

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

Hence, we can say that D $\subset$ A $\subset$ B $\subset$ 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 $\in$ A and A $\in$ B , then x $\in$ B

Answer:

The given statement is false,

example: Let A = { 2,4 }

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

x be 2.

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

But 2 $\notin$ { 1,{2,4},5} i.e. x $\notin$ 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 $\subset$ B and B $\in$ C , then A $\in$ C

Answer:

The given statement is false,

Let , A = {1}

B = { 1,2,3}

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

Here, {1} $\subset$ { 1,2,3} = A $\subset$ B and { 1,2,3} $\in$ {0,{1,2,3},4} = B $\in$ C

But, {1} $\notin$ {0,{1,2,3},4} = A $\notin$ 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 $\subset$ B and B $\subset$ C , then A $\subset$C

Answer:

Let A ⊂ B and B ⊂ C

There is an element x such that

Let, x $\in$ A

$\Rightarrow$ x $\in$ B ( Because A $\subset$ B )

$\Rightarrow$ x $\in$ C ( Because B $\subset$ C )

Hence, the statement is true that A $\subset$ 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 $\not\subset$ B and B $\not\subset$ C , then A $\not\subset$ C

Answer:

The given statement is false

Let , A = {1,2}

B = {3,4,5 }

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

Here, {1,2} $\not\subset$ {3,4,5 } = A $\not\subset$ B and {3,4,5 } $\not\subset$ { 1,2,6,7,8} = B $\not\subset$ C

But , {1,2} $\subset$ { 1,2,6,7,8} = A $\subset$ 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 $\in$ A and A $\not\subset$ B , then x $\in$ B

Answer:

The given statement is false,

Let x be 2

A = { 1,2,3}

B = { 4,5,6,7}

Here, 2 $\in$ { 1,2,3} = x $\in$ A and { 1,2,3} $\not\subset$ { 4,5,6,7} = A $\not\subset$B

But, 2$\notin$ { 4,5,6,7} implies x $\notin$ 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 $\subset$ B and x $\notin$ B , then x $\notin$ A

Answer:

The given statement is true,

Let, A $\subset$ B and x $\notin$ B

Suppose, x $\in$ A

Then, x $\in$ B , which is contradiction to x $\notin$ B

Hence, x $\notin$ A.

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

Answer:

Let A, B, and C be the sets such that A $\cup$ B = A $\cup$ C and A $\cap$ B = A $\cap$ C

To prove: B = C.

A $\cup$ B = A + B - A $\cap$ B = A $\cup$ C = A + C - A $\cap$ C

A + B - A $\cap$ B = A + C - A $\cap$ C

B - A $\cap$ B = C - A $\cap$ C ( since A $\cap$ B = A $\cap$ C )

B = C

Hence proved that B = C.

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

(i) A $\subset$ B(ii) A – B = $\varphi$ (iii) A $\cup$ B = B (iv) A $\cap$ B = A

Answer:

First, we need to show A$\subset$ B $\iff$ A – B = $\varphi$

Let A $\subset$ B

To prove: A – B = $\varphi$

Suppose A – B $\ne$ $\varphi$

this means, x $\in$ A and x $\ne$ B , which is not possible as A $\subset$ B .

SO, A – B = $\varphi$.

Hence, A $\subset$ B $⟹$ A – B = $\varphi$.

Now, let A – B = $\varphi$

To prove: A $\subset$ B

Suppose, x $\in$ A

A – B = $\varphi$ so x $\in$ B

Since, x $\in$ A and x $\in$ B and A – B = $\varphi$ so A $\subset$ B

Hence, A$\subset$ B $\iff$ A – B = $\varphi$.

Let A$\subset$ B

To prove : A $\cup$ B = B

We can say B $\subset$ A $\cup$ B

Suppose, x $\in$ A $\cup$ B

means x $\in$ A or x $\in$ B

If x $\in$ A

since A$\subset$ B so x $\in$ B

Hence, A $\cup$ B = B

and If x $\in$ B then also A $\cup$ B = B.

Now, let A $\cup$ B = B
To prove : A$\subset$ B

Suppose : x $\in$A

A $\subset$ A $\cup$ B so x $\in$ A $\cup$ B

A $\cup$ B = B so x $\in$ B

Hence,A$\subset$ B

ALSO, A$\subset$ B $\iff$ A $\cup$ B = B

NOW, we need to show A $\subset$ B $\iff$ A $\cap$ B = A

Let A $\subset$ B

To prove : A $\cap$ B = A

Suppose : x $\in$ A

We know A $\cap$ B $\subset$ A

x $\in$ A $\cap$ B Also ,A $\subset$ A $\cap$ B

Hence, A $\cap$ B = A

Let A $\cap$ B = A

To prove: A $\subset$ B

Suppose : x $\in$ A

x $\in$ A $\cap$ B ( replacing A by A $\cap$ B )

x $\in$ A and x $\in$ B

$\therefore$ A $\subset$ B

A $\subset$ B $\iff$ A $\cap$ B = A

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

Answer:

Given , A $\subset$ B

To prove: C – B $\subset$ C – A

Let, x $\in$ C - B means x$\in$ C but x$\notin$B

A $\subset$ B so x$\in$ C but x$\notin$A i.e. x $\in$ C - A

Hence, C – B $\subset$ C – A

Question 6: Show that for any sets A and B,

A = ( A $\cap$ B ) $\cup$ ( A – B ) and A $\cup$ ( B – A ) = ( A $\cup$ B )

Answer:

A = ( A $\cap$ B ) $\cup$ ( A – B )

L.H.S = A = Red coloured area

R.H.S = ( A $\cap$ B ) $\cup$ ( A – B )

( A $\cap$ B ) = green coloured

( A – B ) = yellow coloured

( A $\cap$ B ) $\cup$ ( A – B ) = coloured part

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

A $\cup$ ( B – A ) = ( A $\cup$ B )

A = sky blue coloured

( B – A )=pink coloured

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

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

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

Question 7: (i) Using properties of sets, show that

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

Answer:

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

We know that A $\subset$ A

and A $\cap$ B $\subset$ A

A $\cup$ ( A $\cap$ B ) $\subset$ A

and also , A $\subset$ A $\cup$ ( A $\cap$ B )

Hence, A $\cup$ ( A $\cap$ B ) = A

Question 7: (ii) Using properties of sets, show that

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

Answer:

This can be solved as follows

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

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

A $\cap$ ( A $\cup$ B ) = A $\cup$ ( A $\cap$ B ) { A $\cup$ ( A $\cap$ B ) = A proved in 7(i)}

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

Question 8: Show that A $\cap$ B = A $\cap$ C need not imply B = C.

Answer:

Let, A = {0,1,2}

B = {1,2,3}

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

Given, A $\cap$ B = A $\cap$ C

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

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

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

Hence, A $\cap$ B = A $\cap$ C need not imply B = C.

Question 9: Let A and B be sets. If A $\cap$ X $=$B $\cap$ X $=$ $\varphi$and A $\cup$ X $=$ B $\cup$ X for some set X, show that A $=$B.

Answer:

Given, A $\cap$ X $=$B $\cap$ X $=$ $\varphi$ 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) $\cup$ $\varphi$ (A $\cap$ X $=$ $\varphi$)

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

= (B$\cap$A)

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

Hence, A = B

Question 10: 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 $=$ $\varphi$

Answer:

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

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

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 $=$ $\varphi$