NCERT Solutions for Class 11 Maths Chapter 1 Sets Ex 1.4
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Q.1: Find the union of each of the following pairs of sets:
(i). P = {1, 4, 6} and Q = {1, 3, 4}
(ii). X = {a, e, i, o, u} and Y = {x, y, z}
(iii). X = {x: x is a natural number and multiple of 3} and Y = {x: x is a natural number less than 6}
(iv). X = {x: x is a natural number and 1 < x ≤ 5} and Y = {x: x is a natural number and 5 < x < 10}
(v). X = {4, 5, 6} and Y = Φ
Solution:
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Q.2: Let A = {x, y} and B = {x, y, z}
Is A ⊂ B? What is A ∪ B?
Solution:
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Q.3: If X and Y are two sets such that X ⊂ Y, then what is X ∪ Y?
Solution:
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Q.4: If P = {1, 2, 3, 4}, Q = {3, 4, 5, 6}, R = {5, 6, 7, 8} and S = {7, 8, 9, 10}
Find the following:
(i). P ∪ Q
(ii). P ∪ R
(iii). Q ∪ R
(iv). Q ∪ S
(v). P ∪ Q ∪ R
(vi). P ∪ Q ∪ S
(vii). Q ∪ R ∪ S
Solution:
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Q.5: Find the intersection of each of the following pairs of sets:
(i). P = {1, 4, 6} and Q = {1, 3, 4}
(ii). X = {a, e, i, o, u} and Y = {x, y, z}
(iii). X = {x: x is a natural number and multiple of 3} and Y = {x: x is a natural number less than 6}
(iv). X = {x: x is a natural number and 1 < x ≤ 5} and Y = {x: x is a natural number and 5 < x < 10}
(v). X = {4, 5, 6} and Y = Φ
Solution 5:
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Q.6: If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}
Find the following:
(i). A ∩ B
(ii). B ∩ C
(iii). A ∩ C ∩ D
(iv). A ∩ C
(v). B ∩ D
(vi). A ∩ (B ∪ C)
(vii). A ∩ D
(viii). A ∩ (B ∪ D)
(ix). (A ∩ B) ∩ (B ∪ C)
(x). (A∪ D) ∩ (B ∪ C)
Solution:
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Q.7: If A = {y: y is a natural number}, B ={y: y is an even natural number}, C = {y: y is an odd natural number} and D = {y: y is a prime number}
Find the following:
(i). A ∩ B
(ii). A ∩ C
(iii). A ∩ D
(iv). B ∩ C
(v). B ∩ D
(vi). C ∩ D
Solution 7:
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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:
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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
(v). C – A
(vi). D – A
(vii). B – C
(viii). B – D
(ix). C – B
(x). D – B
(xi). C – D
(xii). D – C
Solution 9:
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Q.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:
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Q.11: What is R – Q, if R is the set of real numbers and Q is the set of rational?
Solution:
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Q.12: State whether the following statements are true or false. Give reason.
(i). A = {2, 3, 4, 5} and B = {3, 6} are disjoint sets.
(ii). A = {a, e, i, o, u } and B = {a, b, c, d} are disjoint sets.
(iii). A = {2, 6, 10, 14} and B = {3, 7, 11, 15} are disjoint sets.
(iv). A = {2, 6, 10} and B = {3, 7, 11} are disjoint sets.
Solution:
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NCERT Solutions for Class 11 Maths Chapter 1 Sets Ex 1.4 in Hindi
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प्रश्न 4.
यदि A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} और D = {7, 8, 9, 10}, तो निम्नलिखित ज्ञात कीजिए:
(i) A ∪ B
(ii) A ∪ C
(ii) B ∪ C
(iv) B ∪ D
(v) A ∪ B ∪ C
(vi) A ∪ B ∪D
(vii) B ∪ C ∪ D
हल:
(i) A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}
(ii) A ∪ C = {1, 2, 3, 4} ∪ {5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}
(iii) B ∪ C = {3, 4, 5, 6} ∪ {5, 6, 7, 8} = {3, 4, 5, 6, 7, 8}
(iv) B ∪ D = {3, 4, 5, 6} ∪ {7, 8, 9, 10} = {3,4, 5, 6, 7, 8, 9, 10}
(v) A ∪ B ∪C = ({1, 2, 3, 4} {3, 4, 5, 6}) ∪ {5, 6, 7, 8} = {1, 2, 3, 4, 5, 6} ∪ {5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}.
(vi) 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} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B ∪ C ∪ D = ({3, 4, 5, 6} ∪ {5, 6, 7, 8}) ∪ {7, 8, 9, 10} = {3, 4, 5, 6, 7, 8} ∪ {7, 8, 9, 10) = {3, 4, 5, 6, 7, 8, 9, 10}
प्रश्न 5.
प्रश्न 1 में दिए प्रत्येक समुच्चय युग्म का सर्वनिष्ठ समुच्चय ज्ञात कीजिए:
हल:
(i) X ∩ Y= {1, 3, 5} ∩ {1, 2, 3} = {1, 3}
(ii) A ∩ B = {a, e, i, o, u} ∩ {a,b,c} = {a}.
(iii) A ∩ B = {3, 6, 9 …..} ∩ {1, 2, 3, 4, 5} = {3}.
(iv) A ∩ B = {2, 3, 4, 5, 6} ∩{7, 8, 9} = Φ
(v) A ∩ B = {1, 2, 3} ∩ Φ = 0
प्रश्न 6.
यदि A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} और D = {15, 17}; तो निम्नलिखित ज्ञात कीजिए:
(i) A ∩ B
(ii) B ∩C
(iii) A ∩ C ∩D
(iv) A ∩C
(v) B ∩ D
(vi) Á ∩ (B ∪ C)
(vii) A ∩ D
(viii) A ∩(B ∪D)
(ix) (A ∩ B) ∩ (B ∪ C)
(x) (A ∪ D) ∩ (B ∪ C)
हल:
(i) A ∩ B = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13} = {7, 9, 11}
(ii) B ∩ C = {7, 9, 11, 13} ∩ {11, 13, 15} = {11, 13}
(iii) Á ∩ C ∩ D = ({3, 5, 7, 9, 11} ∩ {11, 13, 15}) ∩ {15, 17} = {11} ∩{15, 17} = Φ
(iv) A ∩ C = {3, 5, 7, 9, 11} ∩ {11, 13, 15} = {11}.
(v) B ∩ D = {7, 9, 11, 13} ∩ {15, 17} = Φ
(vi) 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}.
(vii) A ∩ D = {3, 5, 7, 9, 11} ∩ {15, 17} = Φ
(viii) 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).
(ix) A ∩ B = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13} = {7, 9, 11}
B ∪ C = {7, 9, 11, 13} ∪ {11, 13, 15) = {7, 9, 11, 13, 15).
(A ∩ B) ∩ (B ∪ C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}.
(x) A ∪ D = {3, 5, 7, 9, 11} ∪ {15, 17} = {3, 5, 7, 9, 11, 15, 17}
B ∪ C = {7, 9, 11, 13} ∪ {11, 13, 15} = {7, 9, 11, 13, 15}
(A ∪ D) ∩ (B ∪ C) = {3, 5, 7, 9, 11, 15, 17} ∩ {7, 9, 11, 13, 15} = {7, 9, 11, 15}
प्रश्न 7.
यदि A = {x : x एक प्राकृत संख्या है}, B = {x : x एक सम प्राकृत संख्या है} C = {x : x एक विषम प्राकृत संख्या है}, D = {x : एक अभाज्य संख्या है} तो निम्नलिखित को ज्ञात कीजिए:
(i) A ∩ B
(ii) A ∩C
(iii) A ∩ D
(iv) B ∩C
(v) B ∩D
(vi) C ∩ D
हल:
A = {x : x एक प्राकृत संख्या है} = {1, 2, 3, 4……} B = {x : x एक समं प्राकृत संख्या है} = {2, 4, 6, 8…} C = {x : x एक विषम प्राकृत संख्या है} = {1, 3, 5, 7…} D = {x : x एक अभाज्य संख्या है} = {2, 3, 5, 7, 11….}
(i) A ∩ B = {1, 2, 3, 4….} ∩{2, 4, 6, 8….} = {2, 4, 6, 8….} = B
(ii) A ∩C = {1, 2, 3, 4……} ∩ {1, 3, 5, 7….} = {1, 3, 5, 7….} = C
(iii) A ∩D = {1, 2, 3, 4…} ∩ {2, 3, 5, 7…..} = {2, 3, 5, 7……} = D
(iv) B ∩ C = {2, 4, 6, 8…} ∩ {1, 3, 5, 7……} = Φ
(v) B ∩D = {2, 4, 6, 8…..} ∩ {2, 3, 5, 7…..} = {2}
(vi) C ∩D = {1, 3, 5, 7…..} ∩{2, 3, 5, 7, 11…….} = {3, 5, 7, 11, 13……} = {x : x एक विषम अभाज्य संख्या}
प्रश्न 8.
निम्नलिखित समुच्चय युग्मों में से कौन से युग्म असंयुक्त हैं?
(i) {1, 2, 3, 4} तथा {x : x एक प्राकृत संख्या है और 4 ≤ x ≤ 6}
(ii) {a, e, i, 0, u} तथा {c, d, e, f}
(iii) {x : x एक सम पूर्णांक है। और {x : x एक विषम पूर्णाक है।
हल:
(i) मान लीजिए E = {1, 2, 3, 4} F = {x : x एक प्राकृत संख्या और 4 ≤ x ≤ 6} = {4, 5, 6} अवयव 4, E और F दोनों समुच्चयों में है। अत: दोनों युग्म असंयुक्त नहीं हैं।
(ii) दिये हुए समुच्चयों में अवयव उभयनिष्ठ है। अत: यह असंयुक्त समुच्चय नहीं है।
(iii) मान लीजिए A = {x : x एक सम पूर्णांक हैं। = {….-4, -2, 0, 2, 4…} B = {x : x एक विषम पूर्णांक है} = {….-5, -3, -1, 1, 3, 5…..} A और B समुच्चयों में कोई भी अवयव उभयनिष्ठ नहीं है। अत: यह समुच्चय असंयुक्त है।
प्रश्न 9.
यदि 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}, तो निम्नलिखित को ज्ञात कीजिए:
(i) A – B
(ii) A – C
(iii) A – D
(iv) B – A
(v) C – A
(vi) D – A
(vii) B – C
(viii) B – D
(ix) C – B
(x) D – B
(xi) C – D
(xii) D – C
हल:
(i) A – B = {3, 6, 9, 12, 15, 18, 21} – {4, 8, 12, 16, 20} = {3, 6, 9, 15, 18, 21}
(ii) A – C = {3, 6, 9, 12, 15, 18, 21} – {2, 4, 6, 8, 10, 12, 14, 16} = {3, 9, 15, 18, 21}
(iii) A – D = {3, 6, 9, 12, 15, 18, 21} – {5, 10, 15, 20} = {3, 6, 9, 12, 18, 21}
(iv) B – A = {4, 8, 12, 16, 20} – {3, 6, 9, 12, 15, 18, 21} = {4, 8, 16, 20}
(v) C – A = {2, 4, 6, 8, 10, 12, 14, 16} – {3, 6, 9, 12, 15, 18, 21} = {2, 4, 8, 10, 14, 16}
(vi) D – A = {5, 10, 15, 20} – {3, 6, 9, 12, 15, 18, 21} = {5, 10, 20}
(vii) B – C = {4, 8, 12, 16, 20} – {2, 4, 6, 8, 10, 12, 14, 16} = {20}
(viii) B – D = {4, 8, 12, 16, 20} – {5, 10, 15, 20} = {4, 8, 12, 16}
(ix) C – B = {2, 4, 6, 8, 10, 12, 14, 16} – {4, 8, 12, 16, 20} = {2, 6, 10, 14}
(x) D – B = {5, 10, 15, 20} – {4, 8, 12, 16, 20} = {5, 10, 15}
(xi) C – D = {2, 4, 6, 8, 10, 12, 14, 16} – {5, 10, 15, 20} = {2, 4, 6, 8, 12, 14, 16}
(xii) D – C = {5, 10, 15, 20} – {2, 4, 6, 8, 10, 12, 14, 16} = {5, 15, 20}
प्रश्न 10.
यदि X = {a, b, c, d} औरै Y = {f, b, d, g} तो निम्नलिखित को ज्ञात कीजिए:
(i) X – Y
(ii) Y – X
(iii) X ∩ Y
हल:
(i) X – Y= {a, b, c, d} – {f, b, d, g} = {a, c}
(ii) Y – X = {f, b, d, g} – {a, b, c, d} = {f, g}
(iii) X ∩Y= {a, b, c, d} ∩ {f, b, d, g} = {b, d}
प्रश्न 11.
यदि R वास्तविक संख्याओं और Q परिमेय संख्याओं के समुच्चय हैं, तो R – Q क्या होगा?
हल:
R = {x : x एक वास्तविक संख्या है।} p= {x : x एक परिमेय संख्या है।}
R – Q = {x : x एक अपरिमेय संख्या है।} अत: यह अपरिमेय संख्याओं का समुच्चय है।
प्रश्न 12.
बताइए कि निम्नलिखित कथनों में से प्रत्येक सत्य है या असत्य? अपने उत्तर का औचित्य भी बताइए।
(i) {2, 3, 4, 5} तथा {3, 6} असंयुक्त समुच्चय हैं।
(ii) {a, e, i, 0, u} तथा {a, b, c, 4} असंयुक्त समुच्चय हैं।
(iii) {2, 6, 10, 14} तथा {3, 7, 11, 15} असंयुक्त समुच्चय हैं।
(iv) {2, 6, 10} तथा {3, 7, 11} असंयुक्त समुच्चय हैं।
हल:
(i) यह कथन सत्य नहीं है क्योंकि समुच्चय {2, 3, 4, 5} और {3, 6} में अवयव 3 उभयनिष्ठ है।
(ii) यह कथन सत्य नहीं है क्योंकि समुच्चय {a, e, i, o, u} और {a, b, c, d} में अवयव a उभयनिष्ठ है।
(iii) यह कथन सत्य है क्योंकि समुच्चय {2, 6, 10, 14} और {3, 7, 11, 15} में कोई अवयव उभयनिष्ठ नहीं है। अत: यह समुच्चय असंयुक्त है।
(iv) यह कथन सत्य है क्योंकि समुच्चय {2, 6, 10} और {3, 7, 11} में कोई अवयव उभयनिष्ठ नहीं है। अतः यह समुच्चय असंयुक्त है।
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