Closures of Relations

1
Closures of Relations

• Section 8.4

2
Recap Properties of Relations

• Let R be a relation on set A.
• R is reflexive if (a,a)?R for every element a?A.
• R is symmetric if (b,a)?R whenever (a,b)?R, where
  a,b?A.

3
Recap Properties of Relations

• R is antisymmetric if whenever (a,b)?R and
  (b,a)?R, then a b, where a,b?A.
• Symmetric and antisymmetric are NOT exactly
  opposites
• A relation can have both of these properties
• Or may lack both of them
• R is transitive if whenever (a,b)?R and (b,c)?R,
  then (a,c)?R, where a,b,c?A.

4
Definition of Closure

• 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.
• Properties reflexive, symmetric, and transitive

5
Example Reflexive closure

• A1,2,3
• R(1,1),(1,2),(2,1),(3,2)
• Is R reflexive?? Why?
• What pairs do we need to make it reflexive?
• (2,2), (3,3)
• Reflexive closure of R (1,1),(1,2),(2,1),(3,2)
  ? (2,2),(3,3) is reflexive.

6
Reflexive Closure

• In terms of the digraph representation
• Add loops to all vertices
• In terms of the 0-1 matrix representation
• Put 1s on the diagonal

7
Example Symmetric closure

• A1,2,3
• R (1,1),(1,2),(2,2),(2,3),(3,1),(3,2)
• Is R symmetric?
• What pairs do we need to make it symmetric? (2,1)
  and (1,3)
• Symmetric closure of R (1,1),(1,2),(2,2),(2,3),
  (3,1),(3,2) ? (2,1),(1,3)

8
Symmetric Closure

• Can be constructed by taking the union of a
  relation with its inverse. (See Example 2, pg.
  497)
• In terms of the digraph representation
• Add arcs in the opposite direction
• In terms of the 0-1 matrix representation
• Add 1s to the pairs across the diagonals that
  differ in value

0 0 1 0 0 1 1 1 0
9
Transitive Closure

• In terms of the digraph representation
• If there is a path from a to b, add an arc from a
  to b (can be complicated)
• A path from a to b in the digraph G is a sequence
  of one or more edges starting at a and ending at
  b.