CMSC 56 | Lecture 15: Closures of Relations

Closures of Relations
Lecture 15, CMSC 56
Allyn Joy D. Calcaben

Closure
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.

Consider a relation R:
We want to add edges to
make the relation reflexive
By adding those edges,
we have made a non-
reflexive relation R into
a reflexive relation
This new relation is called the reflexive closure of R
Reflexive Closure

Theorem
The reﬂexive closure of a relation R on A is obtained by adding (a, a) to R for
each a ∈ A.
Reflexive Closure

Theorem
The reﬂexive closure of a relation R on A is obtained by adding (a, a) to R for
each a ∈ A.
The reflexive closure of R is R U ∆,
where ∆ = { (a,a) | a ∈ R } is the diagonal relation on A.
Reflexive Closure

What is the reflexive closure of the relation R = { (a, b) | a < b } on the set of
integers?
Example

integers?
R = { (a, b) | a < b }
Solution

integers?
R = { (a, b) | a < b }
∆ = { (a, a) | a ∈ Z }
Solution

integers?
R = { (a, b) | a < b }
∆ = { (a, a) | a ∈ Z }
The reflexive closure of R is
R U ∆ = { (a, b) | a < b } U { (a, a) | a ∈ Z }
= { (a, b) | a ≤ b }
Solution

Let’s go back to the previous relation R:
We want to add edges to
make the relation symmetric
By adding those edges,
we have made a non-
symmetric relation R into
a symmetric relation
This new relation is called the symmetric closure of R
Symmetric Closure

Definition
Let R be a relation on A. Then R -1 or the inverse of R is the relation
R -1 = { < y,x > | < x, y > ∈ R}
Symmetric Closure

Definition
Let R be a relation on A. Then R -1 or the inverse of R is the relation
R -1 = { < y,x > | < x, y > ∈ R}
NOTE: This relation is sometimes denoted as RT or Rc and called the
converse of R
Symmetric Closure

Theorem
Let R be a relation on A. The symmetric closure of R, denoted s(R), is the
relation R ∪ R−1.
Symmetric Closure

What is the symmetric closure of the relation R = { (a, b) | a > b } on the set
of positive integers?
Example

R = { (a, b) | a > b }
Solution

R = { (a, b) | a > b }
R−1 = { (b, a) | a > b }
Solution

R = { (a, b) | a > b }
R−1 = { (b, a) | a > b }
The symmetric closure of R is the relation
R ∪ R−1 = { (a, b) | a > b } U { (b, a) | a > b }
= { (a, b) | a ≠ b }
Solution

• A path is a sequences of connected edges from vertex a to vertex b
• No path exists from
the noted start location
• A path that starts and
ends at the same
vertex is called a
circuit or cycle
• Must have length ≥1
Start (a)
End (b)
Start (a)
Start (a)
End (b)
Paths in directed graphs

Which of the following are paths in the directed graph shown in the figure:
a,b,e,d; a,e,c,d,b; b,a,c,b,a,a,b; d,c; c,b,a; e,b,a,b,a,b,e?
What are the lengths of those that are paths?
Which of the paths in this list are circuits?
Example
a b c
ed

Solution
a b c
ed

Solution
a b c
ed
a,b,e,d is a path of length 3.

Solution
a b c
ed
a,e,c,d,b is not a path.

Solution
a b c
ed
b,a,c,b,a,a,b is a path of length 6.

Solution
a b c
ed
d,c is a path of length 1.

Solution
a b c
ed
c,b,a is a path of length 2.

Solution
a b c
ed
c,b,a is a path of length 2.
e,b,a,b,a,b,e is a path of length 6.

30
• The length of a path is the number of edges in the path, not the number of
nodes
More on paths . . .

• What is really needed in most
applications is finding the
shortest path between two
vertices
Shortest PathShortest Path

• Informal definition: If there is a path from a to b, then there should be an
edge from a to b in the transitive closure
• First take of a definition:
• In order to find the transitive closure of a relation R, we add an edge from a to c,
when there are edges from a to b and b to c
• But there is a path from 1 to 4 with no edge!
1
2 3
4
R = { (1,2), (2,3), (3,4) } (1,2) & (2,3) (1,3)
(2,3) & (3,4) (2,4)
1
2 3
4
Transitive Closure

• Second take of a definition:
• In order to find the transitive closure of a relation R, we add an edge from a to c,
when there are edges from a to b and b to c
• Repeat this step until no new edges are added to the relation
• We will study different algorithms for determining the transitive closure
• green means added on
the first repeat
• orange means added on
the second repeat
1
2 3
4
Transitive Closure

Theorem:
If A ⊂ B and C ⊂ B, then A ∪ C ⊂ B.
Theorem:
If R ⊂ S and T ⊂ U then R ◦ T ⊂ S ◦ U.
Corollary:
If R ⊂ S then Rn ⊂ Sn
Theorem:
If R is transitive then so is Rn
Transitive Closure

Trick proof:
Show (Rn)2 = (R2)n ⊂ Rn
Theorem:
If Rk = Rj for some j > k, then Rj + m = Rn for some n ≤ j.
Transitive Closure

Definition
The connectivity relation or the star closure of the relation R, denoted R*, is
the set of ordered pairs <a, b> such that there is a path (in R) from a to b:
∞
R*= U Rn
n=1
Transitive Closure

R contains edges between all the nodes reachable via 1 edge
R ◦ R = R2 contains edges between nodes that are reachable via 2 edges in R
R2 ◦ R = R3 contains edges between nodes that are reachable via 3 edges in R
Rn contains edges between nodes that are reachable via n edges in R
Transitive Closure

R* contains edges between nodes that are reachable via any number of
edges (i.e. via any path) in R
Rephrased: R* contains all the edges between nodes a and b when is a path of length
at least 1 between a and b in R
R* is the transitive closure of R
The definition of a transitive closure is that there are edges between any nodes (a,b)
that contain a path between them
Transitive Closure

Theorem
The transitive closure of a relation R equals the connectivity relation R∗.
Transitive Closure

Lemma
Let A be a set with n elements, and let R be a relation on A. If there is a path
of length at least one in R from a to b, then there is such a path with length
not exceeding n. Moreover, when a = b, if there is a path of length at least one
in R from a to b, then there is such a path with length not exceeding n – 1.
Transitive Closure

Theorem
Let MR be the zero-one matrix of the relation R on a set with n elements.
Then the zero-one matrix of the transitive closure R* is:
Transitive Closure
Nodes reachable with
one application of the
relation
two applications of
the relation
n applications of the
relation

Find the zero-one matrix of the transitive closure of the relation R where
Example
1 0 1
MR = 0 1 0
1 1 0

Find the zero-one matrix of the
transitive closure of the relation
R where
Solution
1 0 1
MR = 0 1 0
1 1 0
1 2
3

R where
Solution
1 0 1
MR = 0 1 0
1 1 0
1 1 1
MR2 = 0 1 0
1 1 1
1 2
3

R where
Solution
1 0 1
MR = 0 1 0
1 1 0
1 1 1
MR3 = 0 1 0
1 1 1
1 2
3

R where
Solution
1 1 1
V 0 1 0
1 1 1
1 0 1
MR* = 0 1 0
1 1 0
1 1 1
V 0 1 0
1 1 1
1 1 1
= 0 1 0
1 1 1

oSet
November 12 – December 05
FINAL EXAM
December 11, 2018
12:00PM – 02:00PM
49
Reminders

CMSC 56 | Lecture 15: Closures of Relations

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