Sets Exercise 1.4 Solutions – NCERT Class 11 Mathematics Chapter 1

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

(ii) A = [ a, e, i, o, u} B = {a, b, c}
A ∪ B = {a, b, c, e, i, o, u}

(iii) A = {x : x is a natural number and multiple of 3} B = {x : x is a natural number less than 6}
Upon simplifying sets A and B can be rewritten as
A = {3, 6, 9, 12, 15, ………}
B = {1, 2, 3, 4, 5}
A ∪ B = {1, 2, 3, 4, 5, 6, 9, 12, 15, ………..}

(iv) A = {x : x is a natural number and 1 < x ≤6 } B = {x : x is a natural number and 6 < x < 10 }
Upon simplifying sets A and B can be rewritten as
A = {2, 3, 4, 5, 6}
B = {7, 8, 9}
A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}

(v) A = {1, 2, 3}, B = φ
A ∪ B = {1, 2, 3} φ = {1, 2, 3}


2. Let A = { a, b }, B = {a, b, c}. Is A ⊂ B ? What is A ∪ B ?
Given
A = {a, b}
B = {a, b, c}
Yes A ⊂ B as every element in set A is also in Set B

A ∪ B = {a, b} ∪ {a, b, c} = {a, b, c}
A ∪ B = {a, b, c}


3. If A and B are two sets such that A ⊂ B, then what is A ∪ B ?
Because A is subset of B
Every element of A is an element of B
A ∪ B = B


4. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}and D = {7, 8, 9, 10} Find
(i) A ∪ B

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

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

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

(iv) B ∪ D
B ∪ D = {3, 4, 5, 6} {7, 8, 9, 10} = {3, 4, 5, 6, 7, 8, 9, 10}
B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

(v) A ∪ B ∪ C
A ∪ B ∪ C = {1, 2, 3, 4} ∪ {3, 4, 5, 6} ∪ {5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}
A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

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

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


5. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}and D = {7, 8, 9, 10} Find
(i) A ∩ B

A ∩ B = {1, 2, 3, 4} ∩ {3, 4, 5, 6} = {3, 4}
A ∩ B = {3, 4}

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

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

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

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

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

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


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
A ∩ B = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13} = {7, 9, 11}
A ∩ B = {7, 9, 11}

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

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

(iv) A ∩ C
A ∩ C = {3, 5, 7, 9, 11} ∩ {11, 13, 15} = {11}
A ∩ C = {11}

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

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

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

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

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

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


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}
D = {x : x is a prime number}
Then Find

(i) A ∩ B
A ∩ B = {x : x is a natural number} {x : x is an even natural number}
A ∩ B = {x : x is an even natural number}

(ii) A ∩ C
A ∩ C = {x : x is a natural number} {x : x is an odd natural number}
A ∩ C = {x: x is an odd natural number}

(iii) A ∩ D
A ∩ D = {x : x is a natural number} {x : x is a prime number}
A ∩ D = {x : x is a prime number}

(iv) B ∩ C
B ∩ C = {x : x is an even natural number} ∩ {x : x is an odd natural number}
B ∩ C = φ

(v) B ∩ D
B ∩ D = {x : x is an even natural number} ∩ {x : x is a prime number}
B ∩ D = {2}

(vi) C ∩ D
C ∩ D = {x : x is an odd natural number} ∩ {x : x is a prime number}
C ∩ D = {2}


8. Which of the following pairs of sets are disjoint
(i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6}
Let’s first simplify {x : x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}
So {1, 2, 3, 4} and {4, 5, 6} have nothing in common hence these are disjoint

(ii) {a, e, i, o, u} and {c, d, e, f}
Sets {a, e, i, o, u} and {c, d, e, f} have element e in common hence these two sets are not disjoint

(iii) {x : x is an even integer} and {x : x is an odd integer}
Sets {x : x is an even integer} and {x : x is an odd integer} does not have anything in common that’s why these two disjoint sets.


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}
D = {5, 10, 15, 20} Find

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

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

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

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

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

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

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

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

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

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

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

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


10. If X= {a, b, c, d} and Y = {f, b, d, g} Find
(i) X – Y

X – Y = {a, b, c, d} – {f, b, d, g} = {a, c}
X – Y = {a, c}

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

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


11. If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?
R = Set of Real Numbers = Set of Rational and Irrational Numbers
Q = Set of Rational Numbers
R – Q = Set of Rational and Irrational Numbers – Set of Rational Numbers
R – Q = Set of Irrational Numbers


12. State whether each of the following statement is true or false. Justify your answer.
(i) {2, 3, 4, 5} and {3, 6} are disjoint sets
False

(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets
False

(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets
False

(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets
False


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