Lect1 No 873503264

40
A 2008 A

description

 

Transcript of Lect1 No 873503264

Page 1: Lect1 No 873503264

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Georg Cantor:1845-1918

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PÒa ∈ AL«a´8ÜA�����"ü�8ÜA, B��XJ§�k�Ó���§=

A = B iff (∀x)(x ∈ A ↔ x ∈ B).1 (1)

vk���8Ü���8ܧP�∅. �8Ü´���"£��oº¤

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���m�^S´Ã'�¶~X§{1, 2} = {2, 1}.���­EÑyØK�8Ü�ò"~X§d8Ü���½

§ {1, 2, 2} = {1, 1, 1, 2} = {1, 2}.æ^ù«PÒ�§·�k���±'�Ø@o�ª"~X

�±^{dogs}L«¤k��|¤�8Ü"��4à�¹´{}§L«�8Ü"

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^PÒ{x : P(x)} (½ {x |P(x)})L«÷v^�P�¤ké��8ܶ~X§{x : x ´¢ê}L«¢ê8"{x ∈ A : P(x)}L«®�8ÜA¥¤k÷v^�P���x .{F (x) : x ∈ A}L«éA¥��A^úªF�����¤ké��8¶~X{2x : x ∈ Z}Ò´óê8"{F (x) : P(x)}L«é÷v^�P�é�A^úªF�����¤ké��8ܶ~X§{x2 : x´�ê}L«�ê²��8Ü"

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?�ü�8ÜA, B§·�¡A´B�f8(subset)§eA¥?���Ñ´B¥��"d��¡A�¹uB½B�¹A. ^ÎÒL«�A ⊆ B½B ⊇ A. B�Ø�uBf8¡�B�ýf8"eA´B�ýf8§KP�A ⊂ B.

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�½��8ÜA§·�kA ⊆ A, ∅ ⊆ A.

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-R, Q, Z, N©OL«¢ê8§knê8§�ê8Úg,ê8"@o·�kN ⊂ Z ⊂ Q ⊂ R

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�·��?Ø=�u,�8ÜU�§·�¡U��8"�½�8UÚU�f8A§½ÂA(3U¥)�Ö8AXe

A = {x ∈ U : x 6∈ A},

½=U¥¤kØ3A¥����8"�8U�¤kf8�¤��8Ü{A : A ⊆ U},¡�U��8§P�P(U)½2U .

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�½ü�8ÜA, B§½ÂA, B�

¿8�¤k3A¥½3B¥����8§P�A ∪ B;�8�¤k3A¥�3B¥����8§P�A ∩ B;�é��3A¥�Ø3B¥����8§P�A− B.

A ∪ B = {x : (x ∈ A) or (x ∈ B)};A ∩ B = {x : (x ∈ A) and (x ∈ B)} = {x ∈ A : x ∈ B};A− B = {x : (x ∈ A) and (x 6∈ B)} = {x ∈ A : x 6∈ B}.

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A ∪∅ = AA ∩ U = A

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A ∪ U = UA ∩∅ = ∅ "Æ

A ∪ A = AA ∩ A = A

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A ∪ B = B ∪ AA ∩ B = B ∩ A

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A ∪ A = U ü¥Æ

A ∩ A = ∅ gñÆ

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(A) = A V­Ä½Æ

A ∪ (A ∩ B) = AA ∩ (A ∪ B) = A

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A ∪ (B ∪ C) = (A ∪ B) ∪ CA ∩ (B ∩ C) = (A ∩ B) ∩ C

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A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

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A ∪ B = A ∩ BA ∩ B = A ∪ B

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¡P(U)���f8F�(Uþ�)��8Ü�ê§eF�¹�8U§�é¿ÚÖ$�µ4§=dA, B ∈ F§kA ∪ B, A ∈ F"AO�§ F0 = {∅, U}´��8Ü�ê"

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Outline

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ü�é�a, b�±|¤��kSé§P� (a, b).kSé(a, b), (c, d)�� iff a = c, b = dÓ�¤á"?�ü�8ÜA,B§§��(k�ȽÂXeµ

A× B = {(a, b) : (a ∈ A) and (b ∈ B)}.

?�n�8ÜA1 · · ·An§P§��(k�È�

A1 × A2 × · · · × An

½{P�∏n

i=1 Ai .

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?�8ÜA, B, C§KA×∅ = ∅× A = ∅,eA 6= ∅, B 6= ∅§�A 6= B§KA× B 6= B × A.A× (B × C) = (A× B)× C;A× (B ∪ C) = (A× B) ∪ (A× C);A× (B ∩ C) = (A× B) ∩ (A× C);(B ∪ C)× A = (B × A) ∪ (C × A);(B ∩ C)× A = (B × A) ∩ (C × A).

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é8ÜA, B§lA�B�����'XR´A× B���f8"éA¥�a9B¥�b§¡a, bkR'Xe(a, b) ∈ R. d��PaRb½R(a, b).

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�A = B�§kA�AÏ���'Xµ�'XA× A,�'X∅,ð�'X idA = {(a, a) : a ∈ A}.éA�B���'XR§½ÂR�½Â�Ú��©O�

dom(R) = {x |�3 y ¦� (x , y) ∈ R};ran(R) = {y |�3 x ¦� (x , y) ∈ R}.

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R∼ = {(x , y)|(y , x) ∈ R}.

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R ◦ S = {(x , z)|�3y¦�(xRy)�(ySz)}.

R∼´lB�A���'X§R ◦ S´lA�C���'X"

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Example (Russell & Novig: AIMA, Chapter 5)

Consider the following binary constraint problem PV = {WA, SA, NT , Q, NSW , V , T}U = {red , green, blue}C: no neighboring regions have the same color

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Outline

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8ÜA�8ÜB�'XR¡�lA�B�N�§XJé?¿x ∈ A§�3���y ∈ B¦�(x , y) ∈ R§½=R÷vµ

dom(R) = A,é?¿�x ∈ dom(R)§�3���y ∈ ran(R)§¦�(x , y) ∈ R¤á"

Ï~^��=©i1f , g, h�L«N�"��lA�B�N�f ,�±�¤f : A → B,d�e(x , y) ∈ f ,KPf (x) = y .

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��N�f : A → B¡�÷�§eé?¿b ∈ B�3a ∈ A¦�b = f (a); f¡�ü�§eé?¿ a, a′ ∈ A, f (a) = f (a′)��=�a = a′;ef´÷�q´ü�§K¡f´V�"

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eN�f : A → B���ran(f )´ü:8§K¡f�~�N�¶d�§eran(f ) = {b} ⊆ B§�Pf�cb.�N�

eA = ∅§KlA�?¿8ÜB�k��'X§=�'X∅;ù��'X÷vN��½Â§¡��N�"

ð�N�

ef´lA�A�N�§�é?¿��akf (a) = a,K¡f�ð�N�§P�idA.

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é?¿n ≥ 1,8ÜA���n�$�´��N�N�f : An → A§ùpn�ê"Aþ�0�$�´8ÜA�����"

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�½ü�N�f : A → B, g : B → C,½Â§��EÜN�g ◦ f : A → C�(g ◦ f )(x) = g(f (x)).a

a5¿fÚg��'X�EÜ=´§��EÜN�§�Kþ·�ATP§��'XEÜ�f ◦ g,�S.þ·�P§��N�EÜ�g ◦ f .

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ef , gÑ´÷�£�A/§ü�½V�¤§Kg ◦ f½Xd"eg ◦ f´÷�§Kg½´÷�"eg ◦ f´ü�§Kf½´ü�"eg ◦ f´V�§Kf´ü�§g´÷�"

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Uþ'XR¡�g��(reflexive)§eé?¿x ∈ U·�Ñk(x , x) ∈ R¶�d/§R´g����=�idU ⊆ R.Uþ'XR¡�é¡�§eé?¿x , y ∈ U·�Ñk(x , y) ∈ R��=�(y , x) ∈ R§½=R = R∼.Uþ'XR¡��é¡�§eé?¿x , y ∈ U,d(x , y) ∈ R�(y , x) ∈ R�x = y§½=R ∩ R∼ ⊆ idU .Uþ'XR¡�D4�§eé?¿x , y , z ∈ U,d(x , y) ∈ R�(y , z) ∈ R�(x , z) ∈ R,½=R ◦ R ⊆ R.

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é��8ÜUþ�'XR§XJR´g��§é¡�ÚD4�§K¡R�Uþ��d'X"

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�R´��8ÜUþ����d'X§x ∈ U§P[x ]R�x¤3��da§Ù¥[x ]R = {y ∈ U : xRy}. w,§��8ÜUU�d'XR�±y©�ØÓ��da�¿§ �ØÓ��daØ��"

½Â (û8)

é��8ÜUþ��d'XR§PU/R = {[x ]R : x ∈ U}. ¡�U£�éuR¤�û8"

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eπ´��8ÜU��xf8Ü£=π ⊆ P(U)¤÷v^�µ

π¥���´U���f8ܧπ¥¤k8Ü�¿�uU,π¥?¿ü�8Ü����8§

K¡π´U���y©"

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�½U���y©π§·��±½ÂUþ���'X Rπ¦�U¥ü���a, bk'XRπ iff a, báuπ¥Ó����"�L5§?���8ÜUþ����d'XR, U/R´U��y©§P�πR;dy©πR����d'X=´R.

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�P´'u'X���5�§�é?¿8ÜU, Uþ÷v5�P�'X�3"

éUþ?¿'XR§½ÂR�P-4��÷vP5���¹R����Uþ'X"Pr(R)�R�g�4�§Kr(R) = R ∪ idA.Ps(R)�R�é¡4�§Ks(R) = R ∪ R∼.Pt(R)�R�D44�§K

t(R) = R ∪ R2 ∪ R3 ∪ · · · =∞⋃

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R i .

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a � a;if a � b and b � a then a = b;if a � b and b � c then a � c.

¡� S��8ÜX� S8(partially ordered set, or poset)

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An example of poset

The Hasse diagram of (℘({x , y , z}),⊆)2

2http://en.wikipedia.org/wiki/Hasse_diagram

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Total order and well-order

A partial order �is total (or linear) if for any a, b ∈ X , a � b or b � ais a well-order if every nonempty subset Y of X has a leastelement

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Tree

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A (rooted) tree is a poset (T ,�) such thatT has a unique least element, called the rootthe predecessors of every node are well ordered by �

A path on a tree T is a maximally linearly ordered subset of T .

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Group

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A group is a nonempty set G with a binary operation◦ : G ×G → G such that (a ◦ b) ◦ c = a ◦ (b ◦ c) for alla, b, c ∈ G. An element e in G is called an identity if e ◦x = x ◦efor any x . A semi-group that has an identity is called a monoid.A semi-group with an identity e is a group if each element x hasa unique inverse y such that x ◦ y = y ◦ x = e.

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