Laws of Set Operations [ Mathematical Appendix : Sets ]    

 1.⊂ : (AB  ∧  BC ) ⇒(AC) / φ⊂A⊂Ω / ABAcBc
 2.∪,∩    :idempotent law/communitative law/associative law/distributive law/absorption law/∪Ω,∩Ω
 3.⊂,∪,∩  :A∪B is the smallest set among those which includes both A and B / A∩B is the largest set among those which is included in both A and B /
          A⊂B⇔A∪B=B⇔A∩B=A / A⊂B⇒(A∪C)⊂(B∪C) / A⊂B ⇒ (A∩C)⊂(B∩C)
 4.complement    : the complement of the complement the complement of the empty set/universal set /自らの補集合との∪,∩ / complement and inclusion
 5.empty set 、 6.difference 、 7. de Morgan law
the related pages :Set Notation - Basic symbols, Basic concepts/ power set/対応/写像/特性関数・定義関数/集合族と集合列/被覆/極限集合/集合関数・点関数 

Contents - Sets
Contents 

1.the properties of subsets





transitivity:
 For any set A,B,C,
  ((AB )(BC )) (AC)

* Why?
  transitivity of set inclusion 

【reference】

 Matsuzaka,Introduction to Sets and Topology,Chap1-§1-D-1.4(p.10)








For anysubset of ΩA φAΩ 

* WhyφA

  x (xの元ではないφ) .
  Therefore, x (xφxA).
   [See the truth table of "⇒". ]
  So, φA.
   [See the definition of "⊂". ]

【reference】

 Matsuzaka,Introduction to Sets and Topology,Chap1-§1-D-1.5 (pp.10-11)








For anysubset of ΩA,B,
    (AB) ( Ac Bc )

* Why?
   (A⊃B) ⇔ ( Ac ⊂ Bc )

【reference】

 ・Matsuzaka,Introduction to Sets and Topology,Chap1§1-D-2.15 (p.17)
 ・Nakatani, Logic, 5.3-B-(5.28)(p.123):with proof.
 ・Iyanaga,Sets and TopologyI, Question1.10(vi)(p.23):without proof.   





 


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2.the properties of unions and intersections

a.idempotent law

 For any set A,
  ・AAA 
  ・AAA 

【reference】

 ・Matsuzaka,Introduction to Sets and Topology,Chap1§2-A-2.4 (p.13)

b.Communitative law

 For any set A,B,
  ・AB BA
  ・AB BA 

【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

 For any set A,B,C,
  ・(AB) C A (BC) 
  ・(AB) C A (BC) 

【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

 For any set A,B,C,
  ・A(BC) (AB) (AC) 
  ・A(BC) (AB) (AC) 

【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

 For any set A,B,
  ・A(AB) A
  ・A(AB) A

【reference】

 ・Matsuzaka,Introduction to Sets and Topology,Chap1§2-B-2.11 (p.16);

f.

 For any set A,
  ・Ω A A
  ・Ω A Ω



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3.the properties of the combinations of unions,intersections, and subsets.

a. AB is the smallest set among those which includes both A and B.

 ・A(AB), B(AB)
 ・(AC) (BC (AB)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。

   A∪Bは、A,Bを含む集合のなかで最小。
  → generalization  

b. AB is the largest set among those which is included in both A and B .

A(AB), B(AB)
   "AB is included in both A and B."
・(AC)(BC) (AB)C 
 "AB 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。

   積集合は最大
 → generalization

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):「ABABA」の証明

  AB   ABB    ABA

d.

【reference】


AB  (AC)(BC)  ・Matsuzaka,Introduction to Sets and Topology,Chap1§2-A2.8 (p.14)

AB  (AC) (BC)  ・Matsuzaka,Introduction to Sets and Topology,Chap1§2-B2.8' (p.15)



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4.the properties of the complement 

   (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
   φc  Ω , Ωc φ  
   AAc Ω、 AAc φ
   AB Ac Bc
  * why?
      部分集合と補集合

5.the properties of the empty set.


φA A【reference】 Nakauchi , Logic Workbook ,注意3.1.24(p.144);Matsuzaka,Introduction to Sets and Topology,Chap1§2A2.9 (p.14)

φAφ 【reference】 Nakauchi , Logic Workbook ,注意3.1.24(p.144);Matsuzaka,Introduction to Sets and Topology,Chap1§2B2.9' (p.15)

φcΩ , Ωcφ 【reference】 Matsuzaka,Introduction to Sets and Topology,Chap1§2-D2.12-5 (p.17)

AAcΩ、 AAcφ  【reference】 Matsuzaka,Introduction to Sets and Topology,Chap1§2-D2.12-5 (p.17)

For any set A, φA 
【reference】 Matsuzaka,Introduction to Sets and Topology,Chap1§1-D1.5 (pp.10-11)

 * why φA?
    x (xの元ではないφ) .
    Therefore, x (xφxA). [See the truth table of "⇒". ]
     So, φA. [See the definition of "⊂". ]

6.the properties of the difference


AC (AC)C A(AC)ACc【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)

・(AB)C (AC)(BC) 【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)



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7.de Morgan law


【1】  For anysubset of ΩA,B
          (AB)c = Ac Bc
de Morgan Law
【2】  For anysubset of ΩA,B,
          (AB)c = Ac Bc

【reference】

 ・Nakauchi , Logic Workbook ,theorem3.1.36-7(p.148):証明付  
de Morgan Law


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