14 - for example we use the logical notation loves x,y rather than x,y ∈ loves We sometimes write aR b instead of R a,b for example x loves y or

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Forenames ::= (FirstName, SecondName) Name2 ::= (Forenames, Surname) 3 Relations We wish to capture the concept of objects being related : for example, John loves Mary; 2 < 5 ; two programs P and Q are ‘equal’. Such properties can be expressed in logic using relations on atomic terms, as you have seen in the logic course. Here we deFne relations as special sets. ±or instance, assume the universal set People of all people. We form a set consisting of all ordered pairs of people such that the Frst loves the second: loves = { ( x,y ) : x,y People x loves y } Thus loves People × People . 3.1 Introducing Relations D EFINITION 3.1 (B INARY R ELATIONS ) A binary relation between (arbitrary) sets A and B is a subset of the binary product A × B . We use R,S,. .. to range over relations. If R A 1 × A 2 , we say that R has type A 1 × A 2 . If R A × A , we sometimes just say that R is a binary relation on A . Instead of ( a 1 ,a 2 ) R , we often write R ( a 1 ,a 2 )
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Unformatted text preview: ; for example, we use the logical notation loves ( x,y ) rather than ( x,y ) ∈ loves . We sometimes write aR b instead of R ( a,b ) ; for example, x loves y or 2 < 5 or a ‘ f ‘ b in Haskell. In general, there will be many relations on any set. A relation does not have to be meaningful; any subset of a Cartesian product is a relation. ±or example, for A = { a,b } , there are sixteen relations on A : ∅ { ( a,b ) , ( b,a ) } { ( a,a ) } { ( a,b ) , ( b,b ) } { ( a,b ) } { ( b,a ) , ( b,b ) } { ( b,a ) } { ( a,a ) , ( a,b ) , ( b,a ) } { ( b,b ) } { ( a,a ) , ( a,b ) , ( b,b ) } { ( a,a ) , ( a,b ) } { ( a,a ) , ( b,a ) , ( b,b ) } { ( a,a ) , ( b,a ) } { ( a,b ) , ( b,a ) , ( b,b ) } { ( a,a ) , ( b,b ) } { ( a,a ) , ( a,b ) , ( b,a ) , ( b,b ) } 15...
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This note was uploaded on 01/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

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