Relation_and_Diagrams

Relation_and_Diagrams - Relations and Digraphs Lecture 5...

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Relations and Digraphs Lecture 5 Discrete Mathematical Structures
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Relations and Digraphs Cartesian Product Relations Matrix of Relation Digraph Paths in Digraph
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Ordered Pair and Cartesian Product Ordered pairs: (a,b) (a,b) = (c,d) iff a = c and b = d For any sets A,B A × B = {(a,b)|a A, b B} is called Cartesian Product of A and B Example e {1,2,3} × {a,b} = {(1,a), (2,a), (3,a), (1,b),(2,b), (3,b)} For finite A e B e |A × B|= |A| × |B|
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Generalized Cartesian Product Cartesian product of m nonempty sets: A 1 × A 2 × ... × A m ={(a 1 ,a 2 ,...,a m ) | a i A i ,i=1,2. .., m } What does Cartesian product bring us? Describing the attributes of objects using Cartesian product: A computer program can be characterized by 3 attributes: Language={C(c), Java(j), Fortran(f), Pascal(p), Lisp(l)} Memory={2 meg(2), 4 meg(4), 8 meg(8)} OS={UNIX(u), Windows(w), Linus(l)} Then, any object in Language × Memory × OS can be assigned to a specific program to characterized it.
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Properties of Cartesian Product A × φ = φ× A= φ A × B=B × A A=B A= φ B= φ :Proof Note that for any set S, S × φ = {(x,y)|x S ,y ∈φ }, since no such y exists, so A × φ = φ , and φ× S = φ as well. If A B and A ≠ φ , we can prove that B= φ by contradiction. Assume that A ≠ φ , since A B, let a A, but a B; let b is any element in B(may be in A or not), then (a,b) A × B, but (a,b) B × A, contradiction.
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Strictly speaking: A × (B × C) =? (A × B) × C A × (B × C) = {(a,(b,c))|a A, b B and c C} (A × B) × C = {((a,b),c)|a A, b B and c C} (A C) (B D) A × B C × D Proof: for any (a,b) A × B, a A and b B, so, a C and b D, so, (a,b) C × D However, doesn’t hold. Counterexample: one of A, B is empty set.
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This note was uploaded on 03/31/2010 for the course SE C0229 taught by Professor Tao during the Spring '08 term at Nanjing University.

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Relation_and_Diagrams - Relations and Digraphs Lecture 5...

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