NCERT MOST IMPORTANT QUESTIONS CLASS – 11 | MATHS IMPORTANT QUESTIONS | CHAPTER – 1 | SETS | EDUGROWN |

In This Post we are  providing Chapter-1 SETS NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MOST IMPORTANT QUESTIONS ON SETS

Q1.If A = { 1,2,3,4,5,6}, B = {2,4,6, 8} then find A – B

Ans. We are given the sets A ={ 1,2,3,4,5,6}, B = {2,4,6, 8}

A – B = { 1,2,3,4,5,6}- {2,4,6, 8} = {1, 3, 5}

Q2. Let A and B be two sets containing 3 and 6 elements respectively. Find the maximum and the minimum number of elements in A ∪ B.

Ans.There may be the case when atleast 3 elements are common between both sets

Let a set A = {a, b, c} and B = {a, b, c, d, e, f}

∴ A ∪ B = {a, b, c, d, e, f} implies that the minimum number of elements in A ∪ B are = 6

There may be the case when there are no any elements are common between both sets

lf A = {a, b, c}, B = { d, e, f, g, h, i}

A ∪ B = {a, b, c, d, e, f,g,h,i} implies that the maximum number of elements in A ∪ B are = 9

Q3.If A = {(x,y) : x² + y²= 25  where x, y ∈ W } write a set of all possible ordered pair .

Ans.  We are given the set A = {(x,y) : x² + y²= 25  where x, y ∈ W }

All possible ordered pair of set A are following

For x = 0,y =5, x=3,y=4,for x =4, y =3,for x=5,y =0

A = {(0,5),(3,4),(4,3),(5,0)}

Q4.If A = {1,2,3}, B = {4, 5, 6} and C ={5} verify that A ∪ ( B ∩ C) = (A ∪ B) ∩ A ∪ C.

We are given the sets A = {1,2,3}, B = {4, 5, 6} ,C = {5}

B ∩ C = {5}

LHS

A ∪ ( B ∩ C) = {1,2,3,5}

(A ∪ B) and A ∪ C

A ∪ B = {1,2,3,4,5,6} and A ∪ C = {1,2,3,5}

RHS

(A ∪ B) ∩ A ∪ C = {1,2,3,5}

Therefore

A ∪ ( B ∩ C) = (A ∪ B) ∩ A ∪ C, Hence proved

Q5.From the adjoining Venn diagram, write the value of the following.

(a) A ‘

(b) B’

(c) (A ∩ B)’

Ans. From the venn diagram ,we have

U = {1,2,3,4………15}

A = {7,9.11}. B ={11,12,13,14}

A’ = U – A = {1,2,3,4………15} – {7,9.11} = {1,2,3,4,5,6,8,10,12,13,14,15}

B’ = U – B = {1,2,3,4………15} – {11,12,13,14}= {1,2,3,4,5,6,7,8,9,10,,15}

We have,(A ∩ B) = 11

(A ∩ B)’ = U – (A ∩ B) = {1,2,3,4………15} – {11} = (1,2,3,4,5,6,7,8,9,10,12,13,14,15}

Q6. If P(A) = P(B) show that A = B.

Ans. P(A} and P(B) implies that both are power sets of A and B respectively

Every set is an element of its power set , so A ∈ P(A)

Since, we are given that

P(A) = P(B)

Therefore, A ∈ P(B)

Indicates that every element of A belongs to the set B

So, A ⊂ B……(i)

Similarly B ∈ P(B)

Since, we are given that

P(A) = P(B)

Indicates that every element of B belongs to the set A

So, B ⊂ A……(ii)

From (i) and (ii), we get

A = B, Hence proved

Q7. Let A  and B be sets ; if A∩X = B∩X = ∅ and A∪X = B∪X for some set X. Show that A=B.

Ans. We are given that A∩X = B∩X = ∅ and A∪X = B∪X

To prove   A=B

Proof.  A∪X = B∪X (given)

Multiplying both sides by A∩

A∩ (A∪X) =A∩ (B∪X)

Using distributive property

(A∩ A) ∪ (A ∩ X) = (A∩ B )∪ (A∩ X)

A∩Φ = (A∩ B) ∪ Φ

A = (A∩ B) …………(i)

A∪X = B∪X

Multiplying both sides by B∩

B∩(A∪X) = B∩(B∪X)

(B∩ A) ∪ (B ∩ X) = (B∩ B )∪ (B∩ X)

(B∩ A) ∪φ = B ∪ φ

B = (B∩ A)

B = (A∩ B) …………(ii)

From (i) and (ii)

A = B, Hence proved

Q8.If A ={1,2,3,4,5},then write the proper subsets of A.

Ans. The number of elements in the given sets A ={1,2,3,4,5} are =5

The number of proper subsets of any set are = 2ⁿ – 1

   ^{Where n = number of elements = 5}

  ^{The number of proper subsets of any set are =}  2⁵ – 1 =32 – 1 = 31

Q9.Write the following sets in the Roster form

(i) A={x : x ∈ R, 2x+11 =15}

(ii)B={x |x² =x, x ∈ R}

(iii)C={x = x is a positive factor of the prime number p}

Ans.(i)We have, A={x : x ∈ R, 2x+11 =15}

2x+1= 15 ⇒x= 2

∴ A = {2}

(ii)We have,B={x |x² =x, x ∈ R}

∴ x² = x ⇒ x²-x= 0 ⇒x(x-1)= 0 ⇒x=0,1

∴ B ={0,1}

(iii)We have, C={x = x is a positive factor of the prime number p}

Sice positive factors of a prime u∪mer are 1 ad the number itself,we have

C={1,p}

Q10.For all sets A,B and C show that (A – B) ∩(A – C) = A – (B ∪ C).

Ans. Considering that

x ∈ (A – B)∩(A – C)

⇒ x ∈ (A – B) and x ∈ (A – C)

⇒ (x ∈ A and x∉ B) and (x ∈ A and x∉ C)

⇒ (x ∈ A ) and (x ∉B and  x∉ C )

⇒(x ∈ A ) x∉ (B ∪C)∈

⇒x ∈ A – (B ∪C)

⇒(A – B) ∩(A – C)⊂A – (B ∪ C)…….(i)

Now,Considering that

y ∈A – (B ∪ C)

⇒ y ∈A and y ∉  (B ∪ C)

⇒y ∈A and (y ∉B and y ∉ C)

⇒(y ∈A and y ∉B) and (y ∈A and y ∉ C)

⇒y ∈ (A – B) and y ∈ (A – C)

⇒y ∈ (A – B) ∩ y ∈ (A – C)

⇒A – (B ∪ C) ⊂(A – B) ∩(A – C)………(ii)

From (i) and (ii)

(A – B) ∩(A – C)= A – (B ∪ C), Hence proved

Q11.Let A,B and C be the sets such that A∪B = A∪C and A ∩B = A ∩C,show that B = C.

Ans. According to question, A ∪ B = A ∪ C and A ∩ B = A ∩ C

To show, B = C

Let us assume, x ∈ B So, x ∈ A ∪ B

x ∈ A ∪ C

Hence, x ∈ A or x ∈ C

when x ∈ A, then x ∈ B

∴ x ∈ A ∩ B

As, A ∩ B = A ∩ C

So, x ∈ A ∩ C

∴ x ∈ A or x ∈ C

x ∈ C

∴ B ⊂ C

similarly, it can be shown that C ⊂ B

Hence, B = C

Q12.Show that for any sets A and B

A = (A∩B)∪(A-B) 

Ans.We have to prove

A = (A∩B)∪(A-B)

Taking RHS and solving it

(A∩B)∪(A-B)

Using the property

A -B = A -(A∩B)

A-B = A∩B’

=  (A∩B) ∪ (A∩B’)

Applying distributive property

A∩(B∪C) = (A∩B) ∪(A∩C)

Replacing C = B’ in LHS

A-B = A∩(B∪B’)

= A∩ (U) [since B∪B’ = U]

= A [since A∩ U = A]

= RHS, Hence proved

Q13. Write the following sets in the roster form:

(i) A = {x : x ∈ R, 2x + 15 = 15}

(ii) B = {x : x² = x, x ∈ R}

Ans(i) It is given to us that set

A = {x : x ∈ R, 2x + 11 = 15}

2x + 11 = 15

2x = 15 – 11

2x = 4

x = 2

Therefore in roster form it is written as  A = {2}

(ii) It is given to us that

B = {x : x² =x, x ∈ R}

x² = x

x² – x = 0

x(x – 1) = 0

x = 0, x = 1

Therefore in roster form it is written as

B = {0, 1}

Q14. If  A and B are subsets of the universal set U, then show that 

(i) A ⊂ A ∪ B

Ans. Let’s prove that

A ⊂ A ∪ B

Let  x ∈ A or x ∈ B

If x ∈ A then x ∈ A ∪ B

Hence A ⊂ A ∪ B

Q15. A,B and C are subsets of universal set if A = {2,4,6,8,12,20}, B ={3,6.9.12.15},C ={5,10,15,20} and U is the set of all whole numbers, draw a venn diagram showing the relation of  U,A,B and C.

Ans.