Lect1 No 873503264
description
Transcript of Lect1 No 873503264
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Georg Cantor:1845-1918
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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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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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¡� 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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