18 - inverse of R , written R-1 , is deFned by a R-1 b if...

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3. (Relation Complement) ( a 1 ,a 2 ) R if and only if ( a 1 ,a 2 ) A 1 × A 2 and ( a 1 ,a 2 ) ±∈ R . E XAMPLE 3.5 Let R and S be binary relations on { 1 , 2 , 3 , 4 } such that R = { (1 , 2) , (2 , 3) , (3 , 4) , (4 , 1) } S = { (1 , 2) , (2 , 1) , (3 , 4) , (4 , 3) } We may construct the following relations: R S = { (1 , 2) , (2 , 3) , (3 , 4) , (4 , 1) , (2 , 1) , (4 , 3) } R S = { (1 , 2) , (3 , 4) } R = { (1 , 1) , (1 , 3) , (1 , 4) , (2 , 1) , (2 , 2) , (2 , 4) , (3 , 1) , (3 , 2) , (3 , 3) , (4 , 2) , (4 , 3) , (4 , 4) } We have overloaded notation: R S and R S denotes relation union and intersection respectively when R and S are viewed as relations, and set union and intersection when viewed as sets. Notice that to form a relation union or intersection, the relations must be of the same type. In contrast, we can form the union and intersection of arbitrary sets. It should be clear from the context which interpretation we intend. D EFINITION 3.6 (I DENTITY RELATION ) Given any set S , the identity on A , written id A , is a binary relation on A deFned by id A = { ( a,a ) : a A } . D EFINITION 3.7 (I NVERSE RELATION ) Let R A × B denote an arbitrary binary relation. The
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Unformatted text preview: inverse of R , written R-1 , is deFned by a R-1 b if and only if b R a . Inverse should not be confused with complement: for example, the inverse of is a parent of is is the child of; if z is the cousin of y , then z is in the complement of is a parent of, but not the inverse. Using the R from exam-ple 3.5, the inverse R-1 = { (2 , 1) , (3 , 2) , (4 , 3) , (1 , 4) } . If we take the inverse of the inverse of a relation, we recover the original relation: ( R-1 )-1 = R . D EFINITION 3.8 (C OMPOSITION OF R ELATIONS ) Given R A B and S B C , then the composition of R with S , written R S , is deFned by a R S c if and only if b B. ( a R b b S c ) 19...
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