Composition and Closures of Relations

 

COMPOSITION OF RELATIONS

If R is a relation from set A to set B and S is relation from set B to set C then the composition of R and S denoted by R◦S.

R◦S={(a,c)|there exists some b∈B for which (a,b) ∈R and (b,c) ∈S}

For the relation R◦S, the domain is a subset of R and the range is a subset of S

Example

Let R={(1,1),(1,3),(3,2),(3,4),(4,2)} and S={(2,1),(3,3),(3,4),(4,1)}

R◦S={(1,3),(1,4),((3,1),(4,1)}

S◦R={(2,1),(2,3),(3,2),(3,4),(4,1),(4,3)}

R◦R={(1,1),(1,3),(1,2),(1,4),(3,2)}

S◦S={(3,3),(3,4),(3,1)}

(R◦S)◦R={(1,2),(1,4),(3,1),(3,3),(4,1),(4,3)}

R3=R◦R◦R=(R◦R)◦R=R◦(R◦R)

             ={(1,1),(1,3),(1,2),(1,4)}

 

CLOSURES OF RELATIONS

Definition

The closure of a relation R with respect to property P is the relation obtained by adding the minimum number of ordered pairs to R to obtain property P.

In terms of the digraph representation of R

1. To find the reflexive closure – add loops

2. To find the symmetric closure – add arcs in the opposite direction

3. To find the transitive closure – if there is a path from a to b and b to c, add an arc from a to c

 

Reflexive Closure

Let R be a relation on A.  The reflexive closure of R, denoted r(R) is RU∆A

Add loops to all vertices on the digraph representation of R

Put 1’s on the diagonal elements of the matrix of R

Let A be a set and let ∆A ={(a,a)|a∈R}

∆A is called the diagonal (or the equality) relation on A

Example

If R is the relation on a set A={1,2,3} defined by (a,b)∈R if a<b, where a,b∈A, then            R={(1,2),(1,3),(2,3)}. 

The reflexive closure of R=r(R)=RU∆A

∆A={(1,1), (2,2),(3,3)}

RU∆A ={(1,1), (1,2),(1,3),(2,2),(2,3),(3,3)}

 

Symmetric Closure

Let R be a relation on A. The symmetric closure of R, denoted s(R) is RUR-1

The inverse of R is the relation R-1={(b,a)|(a,b) ∈R}

Note:

To get R-1

    Reverse all the arcs in the digraph representation of R

    Take the transpose MTof the matrix M of R

Example

If R is the relation on a set A={1,2,3} defined by (a,b) ∈ R if a<b, where a,b∈A, then                         R={(1,2),(1,3),(2,3)}. 

The symmetric closure of R=s(R)=RUR-1

R-1={(2,1),(3,1),(3,2)}

RUR-1={(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)}

 

Transitive Closure

The transitive closure of a relation R, t(R), is the smallest transitive relation containing R.

R is transitive iff Rn is contained in R from all n

Hence, if there is a path from x to y and y to z then there must be an arc from x to z, or (x,z) ∈R

The connectivity relation or the star closure of the relation R, denoted R*

 

 

which is equal to R*=RUR2U………URn

Example

Consider the relation R={(1,2),(2,3),(3,3)} on A={1,2,3}.  

R2=R◦R={(1,2),(2,3),(3,3)} ◦ {(1,2),(2,3),(3,3)}

              ={(1,3),(2,3),(3,3)}

R3=R2◦R={(1,3),(2,3),(3,3)}

The transitive closure of a relation R=t(R)= {(1,2),(1,3),(2,3),(3,3)}

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