Chapter 1 समुच्चय (Sets) Ex 1.4 Solutions
Question - 1 : - निम्नलिखित में से प्रत्येक समुच्चय युग्म का सम्मिलन ज्ञात कीजिए:
(i) X = {1, 3, 5}, Y = {1, 2, 3}
(i) A = {a, e, i, 0, u}, B = {a, b, c}
(iii) A = {x : एक प्राकृत संख्या है और 3 का गुणज है।
B = {x : x संख्या 6 से कम एक प्राकृत संख्या है।
(iv) A = {x : x एक प्राकृत संख्या है और 1 < x ≤ 6}
B = {x : x एक प्राकृत संख्या है और 6 < x < 10}
(v) A = {1, 2, 3}, B = Φ
Answer - 1 : -
(i) X ∪ Y = {1, 3, 5} ∪ {1, 2, 3} = {1, 2, 3, 5}
(ii) A ∪ B = {a, e, i, 0, u} ∪ {a, b, c} = {a, b, c, e, i, 0, u}
(iii) A ∪ B = {3, 6, 9….} ∪ {1, 2, 3, 4, 5} = {1, 2, 4, 5 या संख्या 3 का गुणज}
(iv) A = {2, 3, 4, 5, 6}, B = {7, 8, 9}
A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9} अर्थात् {x : 1 < x < 10, x ∈ N}
(v) A ∪B = {1, 2, 3} ∪ Φ = {1, 2, 3}
Question - 2 : - मान लीजिए कि A = {a, b}, B = {a, b, c} क्या A ⊂ B? A ∪ B ज्ञात कीजिए।
Answer - 2 : - A = {a, b}, B = {a, b, c}। समुच्चय A के अवयव a, b समुच्चय B में भी है।
A ⊂ B = A ∪ B = B और A ∪ B = {a, b} ∪ {a, b, c} = {a, b, c}
Question - 3 : - यदि A और B दो ऐसे समुच्चय हैं कि A ⊂ B, तो A ∪ B क्या है?
Answer - 3 : - A ⊂ B समुच्चय A के सभी अवयव समुच्चय B में हैं। A ⊂ B = B.
Question - 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
Answer - 4 : -
(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}
Question - 5 : - प्रश्न 1 में दिए प्रत्येक समुच्चय युग्म का सर्वनिष्ठ समुच्चय ज्ञात कीजिए:
Answer - 5 : -
(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
Question - 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)
Answer - 6 : -
(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}
Question - 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
Answer - 7 : -
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}
Question - 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।
Answer - 8 : -
(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।
Question - 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
Answer - 9 : -
(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}
Question - 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
Answer - 10 : -
(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}