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# slides10 - Relations CMSC 250 1 Definitions q A relation...

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1 CMSC 250 Relations

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2 CMSC 250 Definitions A relation from X to Y is a subset of X × Y If R is a relation and ( x , y ) in R we denote this xRy
3 CMSC 250 Example "Less than" relation from A = {0,1,2} to B = {1,2,3} Use traditional notation 0 < 1, 0 < 2, 0 < 3, 1 < 2, 1 < 3, 2 < 3 1 < 1, 2 < 1, 2 < 2 Or use set notation A × B = {(0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)} R = {(0,1),(0,2),(0,3), (1,2),(1,3), (2,3)} Or use arrow diagrams

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4 CMSC 250 Properties of relations Reflexive Symmetric Transitive ] )[ ( reflexive is xRx A x R 2200 ] )[ , ( symmetric is yRx xRy A y x R 2200 ] )[ , , ( transitive is xRz yRz xRy A z y x R 2200
5 CMSC 250 Equivalence relations Equivalence relations are reflexive, symmetric, and transitive

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6 CMSC 250 Example X = { a,b,c,d,e,f } R = {( a , a ),( b , b ),(c, c ),( d , d ),( e , e ),( f , f ),( a , e ),( a , d ),( d , a ),( d , e ), ( e , a ),( e , d ),( b , f ),( f , b )} Diagram in class
7 CMSC 250 Another Example R is a relation on Z xRy iff x y (mod 5) Diagram in class

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8 CMSC 250
9 CMSC 250 Other properties Reflexive Irreflexive Symmetric Antisymmetric Asymmetric Non-symmetric Transitive ] [ ) ( aRa A a 2200 ] [ ) ( aRa A a 2200

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