Representing B has everything PartialOrders that A has and more: - PowerPoint PPT Presentation

3����13 Mathematics for Computer Science proper subset relation MIT 6.042J/18.062J means A ⊂ B Representing B has everything Partial Orders that A has and more: B ⊄ A Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 rep-po.1 rep-po.2 partial order: properly divides proper subset relation 3� {1,2,3,5,10,15,30} 1� {1,2,5,10} 1 {1,3,5,15} {1,2} � {1,3} {1,5} 3 {1} 1 on {1,2,3,5,10,15,30} Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 rep-po.3 rep-po.4 1

3����13 same shape proper subset {1,2,3,5,10,15,30} {1,2,5,10} {1,3,5,15} as ⊂ example {1,2} {1,3} {1,5} {1} Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 rep-po.5 rep-po.6 same shape partial order: properly divides 3� 1� 1 as ⊂ example ¡5 ¡ � 3 isomorphic on {1,2,3,5,10,15,30} 1 Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 rep-po.7 rep-po.8 �

3����13 Isomorphism Isomorphism All that matters two graphs are isomorphic are the connections: when there is an graphs with the edge-preserving same connections bijection matching matching are isomorphic of their vertices. Albert R Meyer March 22, 2013 rep-po.9 rep-po.10 Albert R Meyer March 19, 2012 Formal Def of Graph Isomorphism p.o. represented by ⊂� G 1 isomorphic to G 2 iff Theorem: Every strict partial order is isomorphic ∃ bijection f:V 1 � V 2 with to a collection of subsets u � v in E 1 IFF f(u) � f(v) in E 2 partially ordered by ⊂ . Albert R Meyer March 22, 2013 rep-po.11 rep-po.12 Albert R Meyer March 19, 2012 3

3����13 p.o. isomorphic to ⊂� p.o. isomorphic to ⊂� proof: map element, a, to proof: map element, a, to the set of elements below it. the set of elements below it. { } { } a maps to a maps to b ∈ A | b Ra OR b = a b ∈ A | b Ra OR b = a { } f(a) :: = R − 1 (a) ∪ a Albert R Meyer March 22, 2013 Albert R Meyer March 22, 2013 rep-po.13 rep-po.14 subsets from divides 30 {1,2,3,5,10,15,30} → � 15 {1,3,5,15} → � 10 {1,2,5,10} → � 2 {1,2} � 5 {1,5} → → � 3 {1,3} → � → 1 { 1 } � Albert R Meyer March 22, 2013 rep-po.15 �

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