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

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online