26-RelsOpsProps

# Matrices are square a a graphs can be drawn with only

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matrices are square |A| × |A| graphs can be drawn with only one set of nodes R = {(1,3), (4,2), (3,3), (3,4)} 0 0 1 0 4 1 1 0 0 3 0 0 0 0 2 0 1 0 0 1 4 3 2 1 1 3 2 4

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Discussion #26 Chapter 5, Sections 3.4-4.5 10/15 Reflexivity Reflexive: 2200 x(xRx) Irreflexive: 2200 x(xRx) = is reflexive is irreflexive is reflexive < is irreflexive “is in same major as” is reflexive “is sibling of” is irreflexive “loves” (unfortunately) is not reflexive, neither is it irreflexive Reflexive Irreflexive Neither 1 3 1 2 1 1 3 2 1 0 3 0 2 0 1 3 2 1 1 3 1 2 0 1 3 2 1
Discussion #26 Chapter 5, Sections 3.4-4.5 11/15 Symmetry Symmetric: 2200 x 2200 y(xRy yRx) Antisymmetric: 2200 x 2200 y(xRy yRx x = y) Asymmetric: 2200 x 2200 y(xRy yRx) “sibling” is symmetric “brother_of” is not symmetric, in general, but symmetric if restricted to males is antisymmetric (if a b and b a, then a=b) < is asymmetric and antisymmetric = is symmetric and antisymmetric is antisymmetric “loves” (unfortunately) is not symmetric, neither is it antisymmetric nor asymmetric

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Discussion #26 Chapter 5, Sections 3.4-4.5 12/15 Symmetry (continued…) symmetric: antisymmetric: Symmetric Antisymmetric (Asymmetric too, if no 1’s on the diagonal) No symmetry properties 1 1 3 1 0 2 1 0 1 3 2 1 (always both ways) (but to self is ok) (never both ways) 0 1 3 1 2 0 1 3 2 1 1 1 3 1 2 0 1 3 2 1 asymmetric: (and never to self) (never both ways)
Discussion #26 Chapter 5, Sections 3.4-4.5 13/15 Transitivity Transitive: 2200 x 2200 y 2200 z(xRy yRz xRz) “taller” is transitive < is transitive x<y<z “ancestor” is transitive “brother-in-law” is not transitive Transitive: “If I can get there in two, then I can get there in one.” (for every x, y, z) if and then x y z

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• Winter '12
• MichaelGoodrich
• Binary relation, Transitive relation, Symmetric relation

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