Tag: NCERT CLASS11 MATHS SOLUTION EXERCISE 1 SETS

NCERT MATHS SOLUTION CLASS 11 SETS CHAPTER 1 EXERCISE 1.4

NCERT MATHS SOLUTION CLASS 11  CHAPTER 1 SETS EXERCISE 1.4-1.6

EXERCISE 1.4

Q.1.(i) Find the union of each of the following pairs of sets: X = {1, 3, 5}, Y = {1, 2, 3}

Solution: Given: X = {1, 3, 5} Y = {1, 2, 3}
∴ X U Y = {1, 2, 3, 5}

Q.1.(ii) Find the union of each of the following pairs of sets: A = {a, e, i, o, u} and B = {a, b, c}

Solution: Given: A = {a, e, i, o, u} and B = {a, b, c}
∴ A U B = {a, b, c, e, i, o, u}

Q.1.(iii) Find the union of each of the following pairs of sets:
A = {x: x is a natural number and multiple of 3} and
B = {x: x is a natural number less than 6}

Solution: Given:
A = {x: x is a natural number and multiple of 3}
⇒ A = {3, 6, 9, 12….} and
B = {x: x is a natural number less than 6}
⇒ B = {1, 2, 3, 4, 5}
∴ A U B = {3, 6, 9, 12….}
= {x: x is a natural number and multiple of 3}

Q.1.(iv) Find the union of each of the following pairs of sets:
 A = {x: x is a natural number and 1 < x ≤ 6} and
 B = {x: x is a natural number and 6 < x < 10}

Solution: Given: A = {x: x is a natural number and 1 < x ≤ 6}
⇒ A = {2, 3, 4, 5, 6} and
B = {x: x is a natural number and 6 < x < 10}
⇒ B = {7, 8, 9}
∴ A U B = {2, 3, 4, 5, 6, 7, 8, 9}
= {x: x is a natural number and 1 < x < 10}

Q.1.(v) Find the union of each of the following pairs of sets: A = {1, 2, 3} and B = Φ

Solution: Given: A = {1, 2, 3} and B = Φ
∴ A U B = {1, 2, 3, 5}

Q.2.  Let A = {a, b} and B = {a, b, c} Is A ⊂ B?
What is A ∪ B?

Solution: Given: A = {a, b} and B = {a, b, c}
Yes,
A ⊂ B
A ∪ B
= {a, b, c}
⇒ B

Q.3. If A and B are two sets such that A ⊂ B, then what is A ∪ B?

Solution: Here A ⊂ B,
∴ A ∪ B = B

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 the following:
(i)  A ∪ B
(ii)  A ∪ C
(iii)  B ∪ C
(iv)  B ∪ D

Solution:
(i) Given: A = {1, 2, 3, 4}, B = {3, 4, 5, 6}
  ∴ A ∪ B
 = {1, 2, 3, 4, 5, 6}

(ii) Given: A = {1, 2, 3, 4}, C = {5, 6, 7, 8}
 ∴ A ∪ C
= {1, 2, 3, 4, 5, 6, 7, 8}

(iii) Given: B = {3, 4, 5, 6}, C = {5, 6, 7, 8}
 ∴ B ∪ C
= {3, 4, 5, 6, 7, 8}

(iv) Given: B = {3, 4, 5, 6} D = {7, 8, 9, 10}
 ∴ B ∪ 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 the following:
(v)  A ∪ B ∪ C
(vi)  A ∪ B ∪ D
(vii)  B ∪ C ∪ D

Solution: Given: A = {1, 2, 3, 4}, B = {3, 4, 5, 6} and C = {5, 6, 7, 8}
∴ A ∪ B ∪ C
= {1, 2, 3, 4, 5, 6, 7, 8}

(vi) Given: A = {1, 2, 3, 4}, B = {3, 4, 5, 6} and D = {7, 8, 9, 10}
∴ A ∪ B ∪ D
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(vii) Given: B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}
∴ B ∪ C ∪ D
= {3, 4, 5, 6, 7, 8, 9, 10}

Q.5. Find the intersection of each pairs of sets of question above.
(i) A ∩ B
(ii) A ∩ C
(iii) B ∩ C
(iv) B ∩ D

Solution:
(i) Given: A = {1, 2, 3, 4}, B = {3, 4, 5, 6},
   ∴ A ∩ B = {3, 4}

(ii) Given: A = {1, 2, 3, 4}, C = {5, 6, 7, 8}
   ∴ A ∩ C  = ϕ

(iii) Given: B = {3, 4, 5, 6}, C = {5, 6, 7, 8}
   ∴ B ∩ C = {5, 6}

(iv) Given: B = {3, 4, 5, 6}, D = {7, 8, 9, 10}
   ∴ B ∩ D = ϕ

Q.5. Find the intersection of each pairs of sets of question above.
(v) A ∩ B ∩ C
(vi) A ∩ B ∩ D
(vii) B ∩ C ∩ D

(v) Given: A = {1, 2, 3, 4}, B = {3, 4, 5, 6} and C = {5, 6, 7, 8}
   ∴ A ∩ B ∩ C = ϕ

(vi) Given: A = {1, 2, 3, 4}, B = {3, 4, 5, 6} and D = {7, 8, 9, 10}
   ∴ A ∩ B ∩ D = ϕ

(vii) Given: B = {3, 4, 5, 6} and C = {5, 6, 7, 8} and D = {7, 8, 9, 10}
   ∴ B ∩ C ∩ D = ϕ

Q.6. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; Find
(i) A ∩ B
(ii) B ∩ C
(iii) A ∩ C ∩ D
(iv) A ∩ C

Solution:
(i) Given: A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}
 ∴ A ∩ B
= {9, 11}

(ii) Given: B = {7, 9, 11, 13}, C = {11, 13, 15}
 ∴ B ∩ C
= {11, 13}

(iii) Given: A = {3, 5, 7, 9, 11}, C = {11, 13, 15}, D = {15, 17}∴ A ∩ C ∩ D
= {11} ∩ {15, 17}
= ϕ

(iv)
∴ A ∩ C
= {11}

Q.6. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; Find
(v) B ∩ D
(vi) A ∩ (B ∪ C)
(vii) A ∩ D

Solution:
(v) Given: B = {7, 9, 11, 13}, D = {15, 17}
∴ B ∩ D
= ϕ

(vi) Given: A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}
    B ∪ C
= {7, 9, 11, 13, 15}
∴ A ∩ (B ∪ C)
= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15}
= {7, 9, 11}

(vii) Given: A = {3, 5, 7, 9, 11}, D = {15, 17}
∴ A ∩ D
= ϕ

Q.6. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; Find
(viii) A ∩ (B ∪ D)
(ix) (A ∩ B) ∩ (B ∪ C)
(ix) (A ∪ D) ∩ (B ∪ C)

Solution:
(viii) Given: A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, D = {15, 17}
   (B ∪ D)
= {7, 9, 11, 13, 15, 17}
∴ A ∩ (B ∪ D)
= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15, 17}
= {7, 9, 11}

(ix) Given: A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}
   (A ∩ B)
= {7, 9, 11, 13}
  (B ∪ C)
= {7, 9, 11, 13, 15}
∴ (A ∩ B) ∩ (B ∪ C)
= {7, 9, 11}

(ix) Given: A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, D = {15, 17}
   (A ∪ D)
= {3, 5, 7, 9, 11, 15, 17}
  (B ∪ C)
{7, 9, 11, 13, 15}
∴ (A ∪ D) ∩ (B ∪ C)
= {7, 9, 11, 15}

Q.7. If A = {x: x is a natural number},
B ={x: x is an even natural number},
C = {x: x is an odd natural number} and
D = {x: x is a prime number}
Find the following:
(i) A ∩ B
(ii) A ∩ C
(iii) A ∩ D

Solution: 

(i) Given, A = {x: x is a natural number}
                   = {1, 2, 3, 4, 5, . . . . . . . },
 B = {x: x is an even natural number}
     ⇒ {2, 4, 6, 8, . . . . . . . }
 ∴ A ∩ B
⇒ {2, 4, 6, 8, . . . . . . . }
= {x: x is an even natural number}
⇒ {2, 4, 6, 8, . . . . . . . }
= B

(ii) Given, A = {x: x is a natural number}
                    = {1, 2, 3, 4, 5, . . . . . . . },
 C = {x: x is an odd natural number}
    = {1, 3, 5, 7, . . . . . . . }
∴ A ∩ C
= {1, 3, 5, 7, . . . . . . . }
⇒ {x: x is an odd natural number}
= C

(iii) Given, A = {x: x is a natural number}
                     = {1, 2, 3, 4, 5, . . . . . . . }
 D = {x: x is a prime number}
     ⇒ {2, 3, 5, 7, . . . . . . . }

∴ A ∩ D
= {2, 3, 5, 7, . . . . . . . }
⇒ {x: x is a prime number}
= D

Q.7. If A = {x: x is a natural number}, B ={x: x is an even natural number}, C = {x: x is an odd natural number} and D = {x: x is a prime number}
Find the following:
(iv) B ∩ C
(v) B ∩ D
(vi) C ∩ D

Solution:
(iv) Given, B = {x: x is an even natural number}
                    = {2, 4, 6, 8, . . . . . . . },
C = {x: x is an odd natural number}
    = {1, 3, 5, 7, . . . . . . . }
∴ B ∩ C = ϕ

(v) Given, B = {x: x is an even natural number}
                    = {2, 4, 6, 8, . . . . . . . }
D = {x: x is a prime number}
   = {2, 3, 5, 7, . . . . . . . }
∴ B ∩ D = {2}

(vi) Given, C = {x: x is an odd natural number}
                     = {1, 3, 5, 7, . . . . . . . }, and
D = {x: x is a prime number}
   = {2, 3, 5, 7, . . . . . . . }
 ∴ C ∩ D
= {3, 5, 7, . . . . . . . }
= {x: x is a odd prime number}

Q.8. Which of the given pairs of sets are disjoint?
(i)  A = {1, 2, 3, 4} and B = {x: x is a natural number and 4 ≤ x ≤ 6}
(ii)  A = {a, e, i, o, u} and B = {c, d, e, f}
(iii)  A = {x: x is an even integer} and B = {x: x is an odd integer}

Solution:
(i)  A = {1, 2, 3, 4}
B = {x: x is a natural number and 4 ≤ x ≤ 6}
   = {4, 5, 6}
∴ A ∩ B = {4}
Hence, A and B are not disjoint.

(ii)  A = {a, e, i, o, u} and B = {c, d, e, f}
∴ A ∩ B = {e}
Hence, A and B are not disjoint.

(iii)  A = {x: x is an even integer}
          = {2, 4, 6, 8, . . . . . . . }
and B = {x: x is an odd integer}
          = {1, 3, 5, 7, . . . . . . . }
∴ A ∩ B = ϕ
Hence, A and B are disjoint.

Q.9. If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20},
 C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}
Find the following:

(i)  A – B
(ii)  A – C
(iii)  A – D
(iv)  B – A

Solution:
(i) Given, A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}
∴ A – B
= {3, 6, 9, 15, 18, 21}

(ii) Given, A = {3, 6, 9, 12, 15, 18, 21}, C = {2, 4, 6, 8, 10, 12, 14, 16}
∴ A – C
= {3, 9, 15, 18, 21}

(iii) Given, A = {3, 6, 9, 12, 15, 18, 21}, D = {5, 10, 15, 20}
∴ A – D
= {3, 6, 9, 12, 18, 21}

(iv) Given, A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}
∴ B – A
= {4, 8, 16, 20},

Q.9. If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20},
C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}
Find the following:
(v)  C – A
(vi) D – A
(vii) B – C
(viii) B – D

Solution:
(v) Given, A = {3, 6, 9, 12, 15, 18, 21}, C = {2, 4, 6, 8, 10, 12, 14, 16}
∴ C – A
= {2, 4, 8, 10, 14, 16}

(vi) Given, A = {3, 6, 9, 12, 15, 18, 21}, D = {5, 10, 15, 20}
∴ D – A
= {5, 10, 20}

(vii) Given, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}
∴ B – C = {20}

(viii) Given, B = {4, 8, 12, 16, 20} and D = {5, 10, 15, 20}
∴ B – D
= {4, 8, 12, 16}

Q.9. If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20},
C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}
Find the following:(v)  C – A
(vi) D – A
(vii) B – C
(viii) B – D

Solution:
(v) Given, A = {3, 6, 9, 12, 15, 18, 21}, C = {2, 4, 6, 8, 10, 12, 14, 16}
∴ C – A
= {2, 4, 8, 10, 14, 16}

(vi) Given, A = {3, 6, 9, 12, 15, 18, 21}, D = {5, 10, 15, 20}
∴ D – A
= {5, 10, 20}

(vii) Given, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}
∴ B – C = {20}

(viii) Given, B = {4, 8, 12, 16, 20} and D = {5, 10, 15, 20}
∴ B – D
= {4, 8, 12, 16}

Q.9. If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20},
     C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}
Find the following:
(ix) C – B
(x) D – B
(xi) C – D
(xii) D – C

Solution:
(ix) Given, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}
∴ C – B
= {2, 6, 10, 14}

(x) Given, B = {4, 8, 12, 16, 20} and D = {5, 10, 15, 20}
∴ D – B
= {5, 10, 15}

(xi) Given, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}
∴ C – D
= {2, 4, 6, 8, 12, 14, 16}

(xii) Given, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}
∴ D – C
= {5, 15, 20}

10: If X = {a, b, c, d} and Y = {f, b, d, g}
Find the following:
(i)  X – Y
(ii)  Y – X
(iii) X ∩ Y

Solution: (i) Given, X = {a, b, c, d} and Y = {f, b, d, g}
∴ X – Y = {a, c}

(ii) Given, X = {a, b, c, d} and Y = {f, b, d, g}
∴ Y – X = {f, g}

(iii) Given, X = {a, b, c, d} and Y = {f, b, d, g}
∴ X ∩ Y = {b, d}

Q.11. If R is the set of real numbers and Q is the set of rational numbers then what is R – Q?

Solution: R is the set of real numbers and Q is the set of rational numbers. ∴ R – Q = Set of irrational numbers.

Q.12: State whether each of the following statements are true or false. Give reason.
(i) {2, 3, 4, 5} and {3, 6} are disjoint sets.
(ii) {a, e, i, o, u } and {a, b, c, d} are disjoint sets.
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.

Solution:
(i) False.
   {2, 3, 4, 5} ∩ {3, 6}
= {3}
∴ {2, 3, 4, 5} and {3, 6} are not disjoint sets.

(ii) False.
    {a, e, i, o, u } ∩ {a, b, c, d}
 = {a}
∴ {a, e, i, o, u } and {a, b, c, d} are not disjoint sets.

(iii) True.
   {2, 6, 10, 14} ∩ {3, 7, 11, 15}
= ϕ
∴ {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.

(iv) True.
   {2, 6, 10} and {3, 7, 11}
= ϕ
∴ {2, 6, 10} and {3, 7, 11} are disjoint sets.

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EXERCISE 1.5

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 U C)’

Solution:
(i) Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}
 ∴ A’ = U – A
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4}
= {5, 6, 7, 8, 9}

(ii) Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, B = {2, 4, 6, 8}
∴ B’= U – B
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}
= {1, 3, 5, 7, 9}

(iii) Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}
 and C = {3, 4, 5, 6}
∴ (A U C) = {1, 2, 3, 4, 5, 6}
∴ (A U C)’
= U – (A U C)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 5, 6}= {7, 8, 9}

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
(iv) ( A U B)’
(v) (A’)’
(vi) (B – C)

Solution:
(iv) Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}
∴ (A U B) = {1, 2, 3, 4, 6, 8}
∴ ( A U B)’
=  {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 6, 8}= {5, 7, 9}

(v) Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}
∴ A’ = U – A
=  {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4}
= {5, 6, 7, 8, 9}
∴ (A’)’= U – A’
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {5, 6, 7, 8, 9}
= {1, 2, 3, 4}

(vi) Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}
∴ (B – C) = {2, 8}
∴ (B – C)’
= U – (B – C)
⇒ {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 8}
= {1, 3, 4, 5, 6, 7, 9}

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}

Solution:
(i) Given, U = {a, b, c, d, e, f, g, h}, A = {a, b, c}
∴ A’ = {a, b, c, d, e, f, g, h} – {a, b, c}
= {d, e, f, g, h}

(ii) Given, U = {a, b, c, d, e, f, g, h}, B = {d, e, f, g}
∴ B’ = {a, b, c, d, e, f, g, h} – {d, e, f, g}
= {a, b, c, h}

(iii) Given, U = {a, b, c, d, e, f, g, h}, C = {a, c, e, g}
∴ C’ = {a, b, c, d, e, f, g, h} – {a, c, e, g}
= {b, d, f, g}

(iv) Given, U = {a, b, c, d, e, f, g, h}, D = {f, g, h, a}
∴ D’ = {a, b, c, d, e, f, g, h} – {f, g, h, a}
= {b, c, d, e}

3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) {x: x is an even natural numbers}.
(ii) {x: x is an odd natural numbers}.
(iii) {x: x is a positive multiple of 3}
(iv) {x: x is a prime number}

Solution:
(i) U = {x: x is a natural numbers}
∴ complements of the set {x: x is an even natural numbers} is
{x: x is an odd natural numbers}.

(ii) U = {x: x is a natural numbers}
∴ complements of the set {x: x is an odd natural numbers} is
{x: x is an even natural numbers}

(iii) U = {x: x is a natural numbers}
∴ complements of the set {x: x is a positive multiple of 3}is
{x: x ∈ N and x is not a multiple of 3}

(iv) U = {x: x is a natural numbers}
∴ complements of the set {x: x is a prime number} is
{x: x ∈ N and x = 1 and x is a composite number}

3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(v) {x: x is a natural number divisible by 3 and 5}
(vi) {x: x is a perfect square}
(vii) {x: x is a perfect cube}
(viii) {x: x + 5 = 8}

Solution:
(v) U = {x: x is a natural numbers}
∴ complements of the set {x: x is a natural number divisible by 3 and 5} is
{x: x ∈ N and x is not divisible by 3 and 5}

(vi) U = {x: x is a natural numbers}
∴ complements of the set {x: x is a perfect square} is
{x: x ∈ N and x is not a perfect square}

(vii) U = {x: x is a natural numbers}
∴ complements of the set {x: x is a perfect cube} is
{x: x ∈ N and x is not a perfect cube}

(viii) U = {x: x is a natural numbers}
        {x: x + 5 = 8}
Given, x + 5 = 8
        ⇒ x = 3
∴ complements of the given set is
{x: x ∈ N and x ≠ 3}

3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(ix) {x: 2x + 5 = 9}
(x) {x: x ≥ 7}
(xi) {x: x ∈ N ; and 2x + 1 > 10}

Solution:
(ix) U = {x: x is a natural numbers}
      {x: 2x + 5 = 9}
Given, 2x + 5 = 9
        ⇒ 2x = 4
        ⇒ x = 2
∴ complements of the given set is
{x: x ∈ N and x ≠ 2}

(x) U = {x: x is a natural numbers}
    {x: x ≥ 7}
∴ complements of the given set is
{x: x ∈ N and x < 7}

(xi) U = {x: x is a natural numbers}
{x: x ∈ N ; and 2x + 1 > 10}
Given, 2x + 1 > 10
        ⇒ 2x > 9
        ⇒ x > ⁹/₂
∴ complements of the given set is
{x: x ∈ N and x ≤ ⁹/₂}

4.(i) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, B = {2, 3, 5, 7} verify that (A U B)’ = A’ ∩ B’

Solution: Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
 A = {2, 4, 6, 8}, B = {2, 3, 5, 7}
    A U B
= {2, 4, 6, 8} U {2, 3, 5, 7}
= {2, 3, 4, 5, 6, 7, 8}
L.H.S.
(A U B)’ = U – (A U B)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 4, 5, 6, 7, 8}
= {1, 9}
A’ = U – A
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}
= {1, 3, 5, 7, 9}

B’ = U – B
= {1, 2, 3, 4, 5, 6, 7, 8, 9}  – {2, 3, 5, 7}
= {1, 4, 6, 8, 9}
R. H.S.
= A’ ∩ B’
⇒ {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9}
= {1, 9} = L.H.S. 
∴ (A U B)’ = A’ ∩ B’  (Verified).

4.(ii) If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, B = {2, 3, 5, 7}
verify that (A ∩ B)’ = A’ U B’

Solution: Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
 A = {2, 4, 6, 8}, B = {2, 3, 5, 7}
    A ∩ B
= {2, 4, 6, 8} ∩ {2, 3, 5, 7}
= {2}
L.H.S.
(A ∩ B)’
= U – (A ∩ B)
⇒ {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2}
= {1, 3, 4, 5, 6, 7, 8, 9}
A’ = U – A
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}
= {1, 3, 5, 7, 9}

B’ = U – B
= {1, 2, 3, 4, 5, 6, 7, 8, 9}  – {2, 3, 5, 7}
= {1, 4, 6, 8, 9}
R. H.S. = A’ U B’
⇒ {1, 3, 5, 7, 9} U {1, 4, 6, 8, 9}
= {1, 3, 4, 5, 6, 7, 8, 9}
= L.H.S. 
∴ (A ∩ B)’ = A’ U B’  (Verified)

5. Draw appropriate Venn diagram for each of the following:
(i) (A U B)’.
(ii) A’ ∩ B’

(i) Solution:

NCERT MATHS SOLUTION CLASS 11  CHAPTER 1 SETS EXERCISE 1.1-1.5

NCERT MATHS SOLUTION CLASS 11  CHAPTER 1 SETS EXERCISE 1.1-1.5

EXERCISE 1.1

1. (i) Which of the following are sets ? Justify your answer.
The collection of all the months of a year beginning with the letter J.

Answer: One can definitely identify a month that begin with the letter J. So The collection of all months of a year begin with the letter J is a well-defined. Hence, this collection is a set.

1. (ii) Which of the following are sets ? Justify your answer.
The collection of ten most talented writers of India.

Answer: As the collection of ten most talented writers of India may vary from person to person. So the collection is not a well-defined collection. Hence, this collection is not a set.

1. (iii) Which of the following are sets ? Justify your answer.
A team of eleven best-cricket batsmen of the world.

Answer: The criteria for determining the best cricket players of the world can vary from person to person. So the team is not a well-defined collection. Hence, this collection is not a set

1. (iv) Which of the following are sets ? Justify your answer.
 The collection of all boys in your class.

Answer: You can definitely identify all boys in your class. So this collection is a well-defined collection. Hence, this collection is a set.

1. (v) Which of the following are sets ? Justify your answer.
The collection of all natural numbers less than 100.

Answer: One can definitely identify a number that belongs to this collection. So The collection of all whole numbers less than 100 is a well-defined collection. Hence, this collection is a set.

1. (vi) Which of the following are sets ? Justify your answer.
A collection of novels written by the writer Munshi Prem Chand.

Answer: A collection of novels written by Munshi Prem Chand is a well-defined collection, because one can definitely identify a book which written by Munshi Prem Chand. Hence, this collection is a set.

1. (vii) Which of the following are sets ? Justify your answer.
The collection of all even integers.

Answer: The collection of all even integers are well-defined set. There are clear criteria for determining which integers belong to the set. So, this collection is a set.

1. (viii) Which of the following are sets ? Justify your answer.
The collection of questions in this Chapter.

Answer: The collection of all questions in this chapter is known. So this collection is a well-defined. Hence, the collection is a set.

1. (ix) Which of the following are sets ? Justify your answer.
A collection of most dangerous animals of the world.

Answer:  The collection of the most dangerous animals of the world is not a well-defined collection because the criteria for determining the dangerousness of an animal can vary from person to person. Hence, this collection is not a set

2. Let A = (1, 2, 3, 4, 5, 6) Insert the appropriate symbols ∈ or ∉ in the blank spaces:
(i)    5 ………. A          (ii)  8 ………. A
(iii)  0 ………. A          (iv) 4 ………. A
(v)   2 ………. A          (vi) 10 ………. A

Answer:
(i)    5 ∈ A
(ii)   8 ∉ A
(iii)  0 ∉ A
(iv)  4 ∈ A
(v)   2 ∈ A
(vi) 10 ∉ A

3. (i) Write the following sets in roster form:
A = {x: x is an integer and – 3 ≤ x < 7}

Solution: Given, A = {x: x is an integer and – 3 ≤ x < 7}
Roster form of set A as
A ={-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

3. (ii) Write the following sets in roster form:
B = {x: x is a natural number less than 6}

Solution: Given, B = {x: x is a natural number less than 6}
Roster form of set B as
B = {1, 2, 3, 4, 5}

3. (iii) Write the following sets in roster form:
C = {x: x is a two-digit natural number such that the sum of its digits is 8}

Solution: Given, C = {x: x is a two-digit natural number such that the sum of its digits is 8}
Roster form of set C as
C = {17, 26, 35, 44, 53, 62, 71, 80}

3. (iv) Write the following sets in roster form:
D = {x: x is a prime number which is divisor of 60)

Solution: Given, D = {x: x is a prime number which is divisor of 60)
60 = 2×2×3×5
Roster form of set D as
D = {2, 3, 5}

3. (v) Write the following sets in roster form:
E = The set of all letters in the word ‘TRIGONOMETRY’

Solution: Given, E = The set of all letters in the word ‘TRIGONOMETRY’
Roster form of set D as
D = {E, G, I, M, N, O, R, T, Y}

3. (vi) Write the following sets in roster form:
F = The set of all letters in the word ‘BETTER’

Solution: Given, F = The set of all letters in the word ‘BETTER’
Roster form of set F as
F = {B, E, R, T}

4. Write the following sets in the set-builder form:
(i) {3, 6, 9, 12}
(ii) {2, 4, 8, 16, 32}

Solution:
(i) {3, 6, 9, 12}
= {1×3, 2×3, 3×3, 4×3}
Set-builder form of the given set is
{x: x = 3n, n ∈ N and 1 ≤ n ≤ 4}

(ii) {2, 4, 8, 16, 32}
= {2¹, 2², 2³, 2⁴, 2⁵}
Set-builder form of the given set is
{x: x = 2ⁿ, n ∈ N and 1 ≤ n ≤ 5}

4. Write the following sets in the set-builder form:
(iii) {5, 25,125, 625}
(iv) {2, 4, 6 , . . .}
(v) {1, 4, 9 , . . . ,100}

Solution:
(iii) {5, 25,125, 625}
= {5¹, 5², 5³, 5⁴}
Set-builder form of the given set is
{x: x = 5ⁿ, n ∈ N and 1 ≤ n ≤ 4}

(iv) {2, 4, 6 ,…}
Set-builder form of the given set is
{x: x is an even natural number}

(v) {1, 4, 9 , . . . ,100}
= {1, 2², 3² ,…,10²}
Set-builder form of the given set is
{x: x = n², n ∈ N and 1 ≤ n ≤ 10}

5. List all the elements of the following sets:
(i) A = {x: x is an odd natural number}
(ii) B = {x: x is an integer, – ¹/₂ < x < ⁹/₂}
(iii) C = {x: x is an integer, x² ≤ 4}

Solution:
(i) A = {x: x is an odd natural number}
        = {1, 3, 5, 7, . . . . . . }

(ii) B = {x: x is an integer, – – ¹/₂ < x < ⁹/₂}
         = {x: x is an integer, – 0.5 < x < 4.5}
         = {0, 1, 2, 3, 4}

(iii) C = {x: x is an integer, x² ≤ 4}
 (-3) ² = 9 ∉ C        (-2)² = 4 ∈ C
 (-1)² = 1 ∈ C           (-0)² = 0 ∈ C
 (1)² = 1 ∈ C             (2)² = 4 ∈ C
 (3)² = 9 ∉ C
∴ C = {-2, -1, 0, 2, 3}

5. List all the elements of the following sets:
(iv) D = {x: x is a letter in the word “LOYAL”}
(v) E = {x: x is a month of a year not having 31 days}
(vi) F = {x: x is a consonant in the English alphabet which precedes k}.

Solution:
(iv) D = {x: x is a letter in the word “LOYAL”}
          = {A, L, O, Y}

(v) E = {x: x is a month of a year not having 31 days}
         = {February, April, June, September, November}

(vi) F = {x: x is a consonant in the English alphabet which precedes k}
         = {b, c,  d, f, g, h, j}

6. Match each of the set on the left in the roster form with the same set on the right described in set-builder form:

(i) {1, 2, 3, 6}               (a) {x: x is a prime number and a divisor of 6}
(ii)(2. 3}                     (b) {x: x is an odd natural number less than 10}
(iii) {M, A, T, H, E, I, C, S} (c) {x: x is natural number and divisor of 6}
(iv) {1, 3, 5, 7, 6}           (d) {x: x is a letter of the word MATHEMATICS}.

Solution:
(i) All the elements of this set are natural number and divisor of 6.
(ii) 2 and 3 are prime number and a divisor of 6
(iii) All the elements of this set are letters of the word ‘MATHEMATICS’.
(iv) All the elements of this set are odd natural number less than 10.

Ans:
(i) {1, 2, 3, 6}                     (c) {x: x is natural number and divisor of 6}
(ii)(2. 3}                             (a) {x: x is a prime number and a divisor of 6)
(iii) {M, A, T, H, E, I, C, S} (d) {x: x is a letter of the word MATHEMATICS).
(iv) {1, 3, 5, 7, 6}               (b) {x: x is an odd natural number less than 10}

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EXERCISE 1.2

1. Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers.Solution:

Solution:
(i) Odd natural numbers are not divisible by 2.
Therefore, this set is empty.

(ii) The only even prime number is 2.
Therefore, this set is not empty.

1. Which of the following are examples of the null set
(iii) { x: x is a natural numbers, x < 5 and x > 7}
(iv) { y: y is a point common to any two parallel lines}

Solution:
(iii) There is not any natural number which is simultaneously less than 5 and greater than 7.
Therefore, this set is empty.

(iv) Parallel lines do not intersects each other’s.
So, there is not any common point,
Therefore, this set is empty set.

2. Which of the following sets are finite or infinite:
 (i) The set of months of a year.
 (ii) {1, 2, 3 , . . .}
 (iii) {1, 2, 3 , . . . 99, 100}

Answer:
(i) There are 12 months of a year.
    So the given set is finite.

(ii) The set represents all natural numbers.
     So the given set is infinite.

(iii) Above set contains 100 (1 to 100) elements.
      So the set is finite.

2. Which of the following sets are finite or infinite:
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99

Answer:
(iv) Positive Integers greater than 100
     = 101, 102, 103, …
  There are infinite positive integers which are greater than 100.
  So the given set is infinite.

(v) The set of prime numbers less than 99 are {2, 3, 5, . . . . . 89, 97}.
     This set has finite number(25) of elements.
     So the given set is infinite.Therefore

3. State whether each of the following set is finite or infinite:
   (i) The set of lines which are parallel to the x-axis.
(ii) The set of letters in the English alphabet.

Answer:
(i) There are infinite lines parallel to x-axis,
    so the set will have infinite elements.
  ∴ The given set is infinite.

(ii) The English alphabet has 26 letters.
∴ the given set is finite.

3. State whether each of the following set is finite or infinite:
(iii) The set of numbers which are multiple of 5.
(iv) The set of animals living on the earth.
(v) The set of circles passing through the origin (0,0)

Answer:
(iii) The set of numbers which are multiple of 5 is {5, 10, 15, . . . . }
∴ the given set is infinite.

(iv) The number of animals living on the earth is quite a large number but they can be counted.
So,the set of animals living on the earth is a finite.

(v) Infinite number of circles can pass through the origin.
So the set will have infinite elements.
Therefore, the given set is infinite

4. (i) In the following, state whether A = B or not.
    A = {a, b, c, d}             B = {d, c, b, a}

Solution: Given, A = {a, b, c, d}
B = {d, c, b, a} = {a, b, c, d}
Every element of A = Every element of B
Also every element of B = Every element of A.
Thus, A and B are equal sets.

4. (ii) In the following, state whether A = B or not.
    A = {4, 8, 12, 16}         B = {8, 4, 16, 18}

Solution: Given, A = {4, 8, 12, 16}
B = {8, 4, 16, 18}
Here 12 ∈ A but 12 ∉ B
∴ A ≠ B

4. (iii) In the following, state whether A = B or not.
     A = {2, 4, 6, 8, 10} 
     B = {x: x is positive even integer and x ≤ 10} 

Solution: Given, A = {2, 4, 6, 8, 10}
B = {x: x is positive even integer and x ≤ 10}
   = {2, 4, 6, 8, 10}
Every element of A = Every element of B
Also every element of B = Every element of A.
Thus, A and B are equal sets.

4. (iv) In the following, state whether A = B or not.
     A = {x: x is a multiple of 10}.
  B = {10, 15, 20, 25, 36 ,…}

Solution: Given, A = {x: x is a multiple of 10}
A = {10, 20, 30,…}
B = {10, 15, 20, 25, 36 ,…}
Here 15 ∈ B but 15 ∉ A
∴ A ≠ B

5.(i) Are the following pair of sets equal? Give reasons.
      A = {2, 3}
      B = {x: x a is solution of x² + 5x + 6 = 0}

[Equal Sets = Two sets A and B are said to be equal if they have exactly the same elements.]

Solution:  A = {2, 3}
B = { x: x a is solution of x² + 5x + 6 = 0}
Given equation :
    x² + 5x + 6 = 0
⇒ x² + 2x + 3x + 6 = 0
⇒ x(x + 2) + 3(x + 2) = 0
or (x + 2)(x + 3) = 0
⇒ x = -2    and    -3
∴ K = {-2, -3}
∴ J ≠ K
So J and K are not equal sets.

5.(ii) Are the following pair of sets equal? Give reasons.
       A = {x: x is a letter in the word FOLLOW}
       B = {y: y is a letter in the word WOLF}

Solution: A = {x: x is a letter in the word FOLLOW}.
    = {F, L, O, W}
 B = {y: y is a letter in the word WOLF}
    = {F, L, O, W}
Every element of A = Every element of B and also every element of B = Every element of A.
Thus, A and B are equal sets.

6. From the sets given below, select equal sets:

A = {2, 4, 8, 12}    B = {1, 2, 3, 4}   
C = {4, 8, 12, 14}   D = {3, 1, 4, 2}
E = {- 1, 1}         F = {0, a}
G = {1, - 1}         H = {0, 1}

Solution:
Clearly elements of B and D are equal though the order is different.
So B = D.
Again elements of E and G are equal though the order is different.
So E = G.
Hence the equal sets are,
   B = D and E = G

EXERCISE 1.3

1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces (i) {2, 3, 4} ……. {1, 2, 3, 4, 5}
(ii) {a, b, c) ……. {b, c, d}
(iii) {x: x ix a student of Class XI of your school} ……. {x: x student of your school}
(iv) {x: x is a circle in the plane} ……. {x: x is a circle in the same plane with radius 1 unit}

Answer:
(i) {2, 3, 4} ⊂ {1, 2, 3, 4, 5}
(ii) {a, b, c) ⊄ {b, c, d}
(iii) {x: x ix a student of Class XI of your school} ⊂ {x: x student of your school}
(iv) {x: x is a circle in the plane} ⊄ {x: x is a circle in the same plane with radius 1 unit}

1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces
(v) {x: x is a triangle in a plane} ……. {x: x is a rectangle in the plane}
(vi) {x: x is an equilateral triangle in a plane} ……. {x: x is a triangle in the same plane
(vii) {x: x is an even natural number} ……. {x: x is an integer}

Ans:
(v) {x: x is a triangle in a plane} ⊄ {x: x is a rectangle in the plane}
(vi) {x: x is an equilateral triangle in a plane} ⊂ {x: x is a triangle in the same plane
(vii) {x: x is an even natural number} ⊂ {x: x is an integer}

2. Examine whether the statements are true or false:
(i) {a, b} ⊄ {b, c, a}
(ii) {a, e} ⊂ {x: x is a vowel in the English alphabet}
(ⅲ) {1, 2, 3} ⊂ {1, 3, 5}

Answer:
(i) False.
    All elements of {a, b} are in {b, c, a}
∴ {a, b} ⊂ {b, c, a}

(ii) True.
   {x: x is a vowel in the English alphabet}
= {a, e, i, o, u}
∴ {a, e} ⊂ {a, e, i, o, u}

(ⅲ) False.
    2 ∈ {1, 2, 3}, but 2 ∉ {1, 3, 5}
∴ {1, 2, 3} ⊄ {1, 3, 5}

2. Examine whether the statements are true or false:
(iv) {a} ⊂ {a, b, c}
(v) {a} ∈ {a, b, c}
(vi) {x: x is an even natural number less than 6} ⊂ {x: x is a natural number which divides 36}

Answer:
(iv) True.
 All elements of {a} are also in {a, b, c}.
 So {a} ⊂ {a, b, c}.

(v) False.
   {a} is not element of {a, b, c}.
∴ {a} ∉ {a, b, c}

(vi) True.
   {x: x is an even natural number less than 6}
= {2, 4} ⊂
   {x: x is a natural number which divides 36}
= {1, 2, 3, 4, 6, 9, 12, 18, 36}
 All elements of {2, 4} are also in {1, 2, 3, 4, 6, 9, 12, 18, 36}.
 So {x: x is an even natural number less than 6} ⊂ {x: x is a natural number which divides 36}.

3. (i)Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?
  {3, 4} ⊂ A

Answer: The statement {3, 4} ⊂ A is incorrect.
{3, 4}is the element of A.
So {3, 4} ∈ A.
∴ {3, 4} ⊄ A.

3. (ii)  Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?   {3, 4} ∈ A

Answer: The statement {3, 4} ∈ A is correct.
 {3, 4}is the element of A.
 So {3, 4} ∈ A,

3. (iii)  Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?   {{3, 4}} ⊂ A

Answer: The statement {{3, 4}} ⊂ A is correct.
{3, 4}is the element of A.
So {3, 4} ∈ A.
Therefore, {{3, 4}} ⊂ A.

3. (iv)  Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?   1 ∈ A

Answer: The statement 1 ∈ A is correct.
1 is the element of A.
 So 1 ∈ A.

3. (v)  Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?   1 ⊂ A 

Answer: The statement 1 ⊂ A is incorrect.
1 is the element of A.
 So 1 ⊄ A.

3. (vi)  Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?   {1, 2, 5} ⊂ A

Answer: The statement {1, 2, 5} ⊂ A is correct.
1, 2, 5 are the element of A.
So, 1, 2, 5 ∈ A.
∴ {1, 2, 5} ⊂ A

3. (vii)  Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?   {1, 2, 5} ∈ A

Answer: The statement {1, 2, 5} ∈ A is incorrect.
  1, 2, 5 are the element of A.
 So, 1, 2, 5 ∈ A.
 Therefore, {1, 2, 5} ∉ A.

3. (viii)  Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?   {1, 2, 3} ⊂ A

Answer: The statement {1, 2, 3} ⊂ A is incorrect.
3 is not an element of A.
 So, 3 ∉ A.
Therefore, {1, 2, 3} ⊄ A

3. (ix)  Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?   ϕ ∈ A

Answer: The statement ϕ ∈ A is incorrect.
  ϕ is not an element of A.
  ϕ is a subset of A.
∴ ϕ ∉ A.

3. (x)  Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?   ϕ ⊂ A.

Answer: The statement ϕ ⊂ A is correct.
  Empty set(ϕ) is a subset of every set.
∴ ϕ is a subset of A.
 Therefore, ϕ ⊂ A.

3. (xi)  Let A = {1, 2, {3, 4}, 5} Which of the following statements are incorrect and why?   {ϕ} ⊂ A

Answer: The statement {ϕ} ⊂ A is incorrect.
 ϕ is not an element of A.
 ∴ {ϕ} ⊄ A

4. Write down all the subsets of the following sets.
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) φ

Answer:
(i) The subsets of {a} are –
φ and {a}

(ii) The subsets of {a, b} are –
φ, {a}, {b} and {a, b}

(iii) The subsets of {1, 2, 3} are –
φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {3, 1}and {1, 2, 3}.

(iv) The subsets of φ are – φ

5. Write the following as intervals.
(i) {x: x ∈ R, – 4 < x ≤ 6}
(ii) {x: x ∈ R, – -12 < x < -10}
(iii) {x: x ∈ R, 0 ≤ x < 7}
(iv)  {x: x ∈ R, 3 ≤ x ≤ 4}

Answer:
(i) {x: x ∈ R, – 4 < x ≤ 6}
 = (-4, 6]

(ii) {x: x ∈ R, -12 < x < -10}
  = (-12, -10)

(iii) {x: x ∈ R, 0 ≤ x < 7}
  = [0, 7)

(iv) {x: x ∈ R, 3 ≤ x ≤ 4}
  = [3, 4]

6. Write the following intervals in set-builder form.
(i) (-3, 0)            (ii) [6, 12] 
(iii) (6, 12]         (iv) [-23. 5)

Answer:
(i)  (-3, 0) = {x: x ∈ R, -3 < x < 0}
(ii) [6, 12] = {x: x ∈ R, 6 ≤ x ≤ 12}

(iii) (6, 12] = {x: x ∈ R, 6 < x ≤ 12}
(iv) [-23, 5) = {x: x ∈ R, -23 ≤ x < 5}

7. What universal set(s) would you propose for each of the following:
 (i) The set of right triangles.
(ii) The set of isosceles triangles.

Answer:
(i) The set of triangles can be the universal set of right triangles.

(ii) The set of triangles can be the universal set of isosceles triangles.

8. Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8} which of the following may be considered as universal set (s) for all the three sets A, B and C
 (i) {0, 1, 2, 3, 4, 5, 6}
(ii) ϕ
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(iv) {1, 2, 3, 4, 5, 6, 7, 8}

Answer: Given A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}
(i) C⊄ {0, 1, 2, 3, 4, 5, 6}
 ∴ {0, 1, 2, 3, 4, 5, 6} is not universal set of A, B and C. 

(ii) Empty set(ϕ) cannot be the universal set for any set.
 So ϕ is not universal set of A, B and C.

(iii) A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10},
  B⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10},
  C⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
∴ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set of A, B and C.

(iv) C⊄ {0, 1, 2, 3, 4, 5, 6, 7, 8}
 ∴ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is not universal set of A, B and C.