Closures on Relations

How do you compute a Closure on a relation?

	add pairs until the relation has a certain property

Can you just add all the possible pairs?

	no, you only add the pairs required for the property


















Reflexive Closure

How do you compute the Reflexive Closure of a relation?

	add (x,x) for each x in the domain

Give the Reflexive Closure of the relation.

	S = {(1,1),(1,2),(2,1)}  defined on {1,2,3}


















Symmetric Closure

How do you compute the Symmetric Closure of a relation?

	add (y,x) for each (x,y) in the relation

Give the Symmetric Closure of the relation.

	R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} defined on {1,2,3,4}

Can you compute the Antisymmetric Closure of a relation?


















Classwork

You may work with a partner.

Give the Symmetric Closure of the relation.

	T = {(2,1),(2,3),(3,1),(3,4),(4,1),(4,3)} defined on {1,2,3,4}


















Transitive Closure

How do you compute the Transitive Closure of a relation?

	add (x,z) for each (x,y) and (y,z) in the relation
	(repeat until no change)

Give the Transitive Closure of the relation.

	R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} defined on {1,2,3,4}


















Classwork

You may work with a partner.

Give the Transitive Closure of the relation.

	T = {(2,1),(2,3),(3,1),(3,4),(4,1),(4,3)} defined on {1,2,3,4}


















Reflexive Transitive Closure

How do you compute the Reflexive Transitive Closure of a relation?

	add both reflexive and transitive pairs

Give the Reflexive Transitive Closure of the relation.

	R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} defined on {1,2,3,4}


















Classwork

You may work with a partner.

Give the Reflexive Transitive Closure of the relation.

	T = {(2,1),(2,3),(3,1),(3,4),(4,1),(4,3)} defined on {1,2,3,4}