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→ Laws of Set Operations : contents → Mathematical Appendix : sets → Mathematical Appendixes |
a.idempotent law | ||
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For any set A, ・A∪A=A ・A∩A=A |
【reference】・Matsuzaka,Introduction to Sets and Topology,Chap1§2-A-2.4 (p.13) |
b.Communitative law |
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For any set A,B, ・A∪B = B∪A ・A∩B = B∩A |
【reference】・Matsuzaka,Introduction to Sets and Topology,Chap1§2-A-2.5 (p.13);・Nakauchi , Logic Workbook ,theorem3.1.9(p.139); |
c.Associative law |
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For any set A,B,C, ・(A∪B) ∪ C = A ∪ (B∪C) ・(A∩B) ∩ C = A ∩ (B∩C) |
【reference】・Matsuzaka,Introduction to Sets and Topology,Chap1§2-A-2.6 (p.13);・Nakauchi, Logic Workbook ,theorem3.1.9(p.139); ・Cramér, Mathematical Methods of Statistics, 1.3(p.6) |
d.Distributive law |
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For any set A,B,C, ・A∪(B∩C) = (A∪B) ∩ (A∪C) ・A∩(B∪C) = (A∩B) ∪ (A∩C) |
【reference】・Matsuzaka,Introduction to Sets and Topology,Chap1§2-B-2.10 (p.15);・Nakauchi , Logic Workbook ,theorem3.1.9(p.139); ・Cramér, Mathematical Methods of Statistics, 1.3(p.5) |
e.Absorption law |
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For any set A,B, ・A∪(A∩B) = A ・A∩(A∪B) = A |
【reference】・Matsuzaka,Introduction to Sets and Topology,Chap1§2-B-2.11 (p.16); |
f. | ||
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For any set A, ・Ω ∩ A = A ・Ω ∪ A = Ω |
→ Laws of Set Operations : contents → Mathematical Appendix : sets → Mathematical Appendixes |
a. A∪B is the smallest set among those which includes both A and B. | ||
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・A⊂(A∪B), B⊂(A∪B) ・(A⊂C) ∧ (B⊂C) ⇒ (A∪B)⊂C |
【reference】・Matsuzaka,Introduction to Sets and Topology,Chap1§2-A-2.2-3;2.7-9 (pp.13-4);・Nakatani, Logic,5.3-B-(5.29)(p.123):証明付。 |
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b. A∩B is the largest set among those which is included in both A and B . | ||
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・A⊃(A∩B), B⊃(A∩B) "A∩B is included in both A and B." ・(A⊃C)∧(B⊃C) ⇒ (A∩B)⊃C "A∩B includes all of the sets which are included in both A and B." |
【reference】・Matsuzaka,Introduction to Sets and Topology,Chap1§2-B-2.2'-3' (p.15);・Nakatani, Logic, 5.3-B-(5.2.31)(p.124):証明略。 |
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c. | 【reference】・Matsuzaka,Introduction to Sets and Topology,Chap1§2-A2.7(p.14);§2-B2.7'(p.15);・Nakauchi , Logic Workbook ,例題3.1.20(p.141); ・Nakatani, Logic, 5.3-B-(5.2.30)(p.123):証明付(p.124);(5.2.32)(p.124):証明略。 ・永倉・宮岡『解析演習ハンドブック[1変数関数編]』1.1.3-(iv) (p.2);演習問題1.1-ex1.1.2(ii)(p.7):「A⊂B⇔A∩B=A」の証明 |
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A⊂B ⇔ A∪B=B ⇔ A∩B=A |
d. |
【reference】 |
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・A⊂B ⇒ (A∪C)⊂(B∪C) | ・Matsuzaka,Introduction to Sets and Topology,Chap1§2-A2.8 (p.14) |
・A⊂B ⇒ (A∩C) ⊂ (B∩C) | ・Matsuzaka,Introduction to Sets and Topology,Chap1§2-B2.8' (p.15) |
→ Laws of Set Operations : contents → Mathematical Appendix : sets → Mathematical Appendixes |
(Ac )c = A |
【reference】・Nakauchi , Logic Workbook ,注意3.1.34(p.147);・Matsuzaka,Introduction to Sets and Topology,Chap1§2-D-2.12-5 (p.17) 包含関係と補集合 ・Nakatani, Logic, 5.3-B-(5.28)(p.123):証明付。 ・永倉・宮岡『解析演習ハンドブック[1変数関数編]』演習問題1.1-ex1.1.4(iii)(p.8):証明 ・Iyanaga,Sets and TopologyI, 問題1.10(vi)(p.23):証明なし |
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φc = Ω , Ωc = φ |
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A∪Ac = Ω、 A∩Ac = φ | ||
A⊃B ⇔ Ac ⊂ Bc * why? |
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・φ∪A =A | 【reference】 Nakauchi , Logic Workbook ,注意3.1.24(p.144);Matsuzaka,Introduction to Sets and Topology,Chap1§2A2.9 (p.14) |
・φ∩A=φ | 【reference】 Nakauchi , Logic Workbook ,注意3.1.24(p.144);Matsuzaka,Introduction to Sets and Topology,Chap1§2B2.9' (p.15) | |
・φc=Ω , Ωc=φ | 【reference】 Matsuzaka,Introduction to Sets and Topology,Chap1§2-D2.12-5 (p.17) | |
・A∪Ac=Ω、 A∩Ac=φ | 【reference】 Matsuzaka,Introduction to Sets and Topology,Chap1§2-D2.12-5 (p.17) | |
・For any set A, φ⊂A |
【reference】 Matsuzaka,Introduction to Sets and Topology,Chap1§1-D1.5 (pp.10-11) | |
* why φ⊂A? ∀ x (xφ) . Therefore, ∀ x (x∈φ⇒x∈A). [See the truth table of "⇒". ] So, φ⊂A. [See the definition of "⊂". ] |
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・A−C = (A∪C)−C = A−(A∩C)=A∩Cc | 【reference】 Nakauchi , Logic Workbook ,theorem3.1.28(pp.144-5):証明付; Matsuzaka,Introduction to Sets and Topology,Chap1§2問題3(a) |
・(A−B)∩C = (A∩C)−(B∩C) | 【reference】 Nakauchi , Logic Workbook ,theorem3.1.28(pp.144-5):証明付; Matsuzaka,Introduction to Sets and Topology,Chap1§2問題4(e) |
→ Laws of Set Operations : contents → Mathematical Appendix : sets → Mathematical Appendixes |
【1】 For any 《subset of Ω》A,B, (A∩B)c = Ac ∪ Bc |
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【2】 For any 《subset of Ω》A,B, (A∪B)c = Ac ∩ Bc 【reference】・Nakauchi , Logic Workbook ,theorem3.1.36-7(p.148):証明付 |
→ Laws of Set Operations : contents → Mathematical Appendix : sets → Mathematical Appendixes |