8.3 Representing Relations
• Directed Graphs
– Vertex
– Arc (directed edge)
– Initial vertex
– Terminal vertex
Example
• Draw the “divides” relation on the set
{2,3,4,5,6,7,8,9} as a directed graph
The zero-one Matrix
Representation MR
• MR is just a zero-one version of the “chart”
representation of R.
2
3
4
5
6
7
8
9
2
1
0
0
0
0
0
0
0
3
0
1
0
0
0
0
0
0
4
1
0
1
0
0
0
0
0
5
0
0
0
1
0
0
0
0
6
1
1
0
0
1
0
0
0
7
0
0
0
0
0
1
0
0
8
1
0
1
0
0
0
1
0
9
0
1
0
0
0
0
0
1
Reflexivity
Directed graph picture
Zero-one matrix picture
Symmetry
Directed graph picture
Zero-one matrix picture
Antisymmetry
Directed graph picture
Zero-one matrix picture
Transitivity
Directed graph picture
Zero-one matrix picture
Theorem
For relations R1 and R2 on set A,
M R1  R2  M R1  M R2 ,
M R1  R2  M R1  M R2 , and
M R1  R2  M R2
M R1
Example: Let 𝑅1 = { 1,2 , 2,1 , 2,2 , 3,3 } and 𝑅2 =
{ 1,1 , 1,2 , 1,3 , 3,2 } be binary relations. Find 𝑀𝑅1 and 𝑀𝑅2 . Use them to
find 𝑀𝑅1∘𝑅2 . Verify by calculating 𝑅1 ∘ 𝑅2 without matrices.
Corollary
For a relation R on set A,
M R n  M R 
[n ]
for any positive integer n.
1 0 1
1 1 0
Example: Let 𝑀𝑅 = 0 0 1 and 𝑀𝑆 = 0 0 0 . Calculate 𝑀𝑅𝑛 and
1 1 0
1 1 1
𝑀𝑆 𝑛 to determine if the relations 𝑅 and 𝑆 are transitive.
8.4 Closures of Relations
• Reflexive closure
• Symmetric closure
Example: Let R be the relation on the set {1,2,3,4} containing the pairs
{ 1,1 , 1,2 , 2,4 , 3,1 , 4,2 }
What is the reflexive closure of R? What is the symmetric closure of R?
Paths in Directed Graphs
• A path in a directed graph is a sequence of
vertices for which any two consecutive
vertices ai and ai+1 in the sequence are
joined by an arc from ai to ai+1.
• Theorem: Let R be a relation on set A, and
n a positive integer. Then there is a path of
length n from a to b in R if and only if (a,b)
is in Rn.
Example:
The “Connectivity Relation” R*
• Let R be a relation on
set A. We define

R R
*
i 1
i
Example: Let 𝑅 be the relation on the set of all people in the world that contains
(𝑎, 𝑏) if 𝑎 has met 𝑏. What is 𝑅𝑛 , where 𝑛 is a positive integer greater than one?
What is 𝑅∗ ?
The Transitive Closure
• For a relation R on a set A, we define the
transitive closure of R to be the smallest
transitive relation containing R.
• Theorem:
Finding transitive closure the “hard” way:
𝑅 = { 1,1 , 1,2 , 2,4 , 3,1 , 4,2 }
Computing R*
• If A is a set with n elements, and R is a
relation on A, then any time there is a path of
length one or more from a to b in R then
there is a path of length n or less.
• So actually
n
R R
*
i
i 1
and
M R* 
M
n
i 1
[i ]
R
• Interestingly, this is not the best way of
computing R*
𝑅 = { 1,1 , 1,2 , 2,4 , 3,1 , 4,2 }
Computing transitive closure the better way:
1
𝑀𝑅 = 0
1
0
⋁
1
0
0
1
0
0
0
0
𝑀𝑅
[3]
0
1
0
0
⋁
1
= 0
1
0
1
0
1
1
𝑀𝑅
0
0
0
0
[2]
1
= 0
1
0
1
1
1
0
1
= 𝑀𝑅 ∗ = 0
1
0
1
1
1
0
0
0
0
0
⋁
1
1
1
1
1
0
0
1
𝑀𝑅
0
0
0
0
1
1
1
1
[4]
1
= 0
1
0
1
1
1
0
0
0
0
0
1
0
1
1
Warshall’s Algorithm
procedure Warshall(MR: n by n zero-one matrix)
W := MR
for k:=1 to n
for i:=1 to n
for j:=1 to n
wij := wij  (wik  wkj)
{ W now contains MR* }
Illustration of Warshall’s Algorithm
1
𝑀𝑅 = 0
1
0
1
0
0
1
0
0
0
0
0
1
0
0
1
0
1
0
1
0
0
1
0
0
0
0
1
0
1
0
0
1
0
1
0
1
0
1
0
1
0
0
0
0
1
1
0
0
1
0
1
0
1
1
0
1
1
0
0
0
0
1
1
0
0
1
0
1
0
1
1
1
1
0
0
0
0
1
1
0
1
0
1
0
1
0
1
1
1
1
0
0
0
0
1
1
1
0
1
8.5 Equivalence Relations
• Definition: A relation R on set A is an
equivalence relation if …
Examples
• aRb if and only if a and b have the same first name (on
the set of students in this class)
• aRb if and only if a ≡ b (mod 5) (on the set of integers)
Equivalence Classes
• If R is a relation on set A, and a is an
element of A, then…
Examples (continued)
• [Michael]
• [4]5
Theorem
For an equivalence relation R on set A and
elements a and b of A, the following are all
logically equivalent:
a) a R b
b) [a]R = [b]R
c) [a]R  [b]R  
Partitions
•
For a set S, a partition of S is a collection
 = {A1, A2, …, Am} of nonempty subsets
of S which satisfies the following
properties:
– Every element of A is in one of the sets Ai.
– For all i, j  {1, 2, …, m},
if i  j then Ai  Aj = 
•
Terminology: We say that the collection
 partitions S.
Theorem
• Let R be an equivalence relation on set S.
Then the equivalence classes of R partition
S. Conversely, for any partition  of S there
is an equivalence relation R whose
equivalence classes are the sets in .
Visual
…,-10,
…,
,-9,
,
, ,-7,
,-8,
, ,
,-6,
, ,
,-5,
,-4,
,-3,
, ,
,-2,
,-1,
,
,0,
,
, ,
,1,
,2,
,3,
,
, ,
,4,
,5,
,6,
,
, ,
,7,
,8,
,9,
,
, , …
,10,
[2]={… ,-10, -7, -4, -1, 2, 5, 8, …}
[0]={… ,-9, -6, -3, 0, 3, 6, 9, …}
[1]={… ,-8, -5, -2, 1, 4, 7, 10, …}
Visual
[2]={… ,-10, -7, -4, -1, 2, 5, 8, …}
[0]={… ,-9, -6, -3, 0, 3, 6, 9, …}
[1]={… ,-8, -5, -2, 1, 4, 7, 10, …}
…,-10,
,
…,
,-9,
…,
,
,-7,
,
,-8,
,
,-6,
,
,-4,
,
,-5,
,
,-3,
,
,-1, , ,2, , ,5, , ,8, ,
, …
,
, …
,-2,
,0, , ,3, , ,6, , ,9,
, ,1, , ,4, , ,7, , ,10, …
Example: Turning a partition into an
equivalence relation
𝐴1 = 1,5 ,
𝐴2 = 2,3,6 ,
𝐴3 = 4 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑆 = {1,2,3,4,5,6}
Zero-One Matrix Representation of
an Equivalence Relation Examples
Digraph Representation of an
Equivalence Relation Examples
Equivalence as “sameness”
• Almost every equivalence relation
definition comes down to identifying some
notion of “sameness”
– Same remainder when divided by n
– Same name
– Same set of a partition
Number of Partitions of a Set with n
Elements
• n=1
• n=2
• n=3
• n=4
Recurrence Relation for the Number of
Partitions of a Set with n Elements
8.6 Partial Orderings
Let A be a set, and R a relation on A. We say
that R is a partial ordering if and only if…
In this case we say that the pair (A, R) is a
partially ordered set (poset).
Examples:
1. The real numbers R under the  relation
2. The real numbers R under the  relation
3. The positive integers under the “divides”
relation
4. Any set of sets under the  (subset)
relation
5. The cartesian product ZZ under the
“(,)” relation R. (i.e. (x,y) R (z,w) if and
only if x  z and y  w.)
Convention
• The symbol ≼ is the default symbol used
to represent a partial ordering.
• Example: “Let A be a set, and let ≼ be a
partial ordering on A.”
Comparable and Incomparable
Elements
• Two elements a and b of a partially
ordered set are said to be incomparable if
and only if the statements a ≼ b and b ≽ a
are both false. Otherwise the elements are
comparable.
• Examples:
– Subsets
– Cartesian products
– Divides relation
Examples:
Total Orderings
• Let A be a set, and let ≼ be a partial ordering on
A. We say that ≼ is a total ordering provided…
In this case we say that the pair (A, ≼) is a totally
ordered set. (linearly ordered set, chain)
• Examples:
1) Real numbers under 
2) Any set of strings under the
“dictionary”, or lexicographic,
ordering
A partial order on 𝐴1 × 𝐴2 “induced” by
partial orders ≼1 on 𝐴1 and ≼2 on 𝐴2
• Lexicographic ordering ≼
• Example:
Well-Ordered Sets
• A set S is well-ordered by the partial
ordering ≼ if and only if every nonempty
subset of S has a least element (minimum
element).
• Examples: ℕ, ℤ+
•
Non-Examples: ℝ+ ,
ℚ+ , ℤ
Hasse Diagrams
• Begin with the digraph representation of
the partial ordering
• Omit the reflexive loops
• Omit all edges which would be implied by
transitivity
• Orient all vertices and arcs so that the
direction of each arc is up.
• Remove the direction arrow from each arc
Hasse Diagram Example I
• Pairs in {1,2,3}{1,2,3} under lexicographic order
Hasse Diagram Example I
• Pairs in {1,2,3}{1,2,3} under lexicographic order
Hasse Diagram Example I
• Pairs in {1,2,3}{1,2,3} under lexicographic order
Hasse Diagram Example I
• Pairs in {1,2,3}{1,2,3} under lexicographic order
Hasse Diagram Example I
• Pairs in {1,2,3}{1,2,3} under lexicographic order
Hasse Diagram Example II
• Integers 1-12 under “divides”
Hasse Diagram Example III
• Integers 1-12 under “divides”
Maximal and Minimal Elements
Given a poset (𝑆, ≼) and an element 𝑎 ∈ 𝑆, we say
• 𝑎 is maximal in the poset if
• 𝑎 is minimal in the poset if
• 𝑎 is the greatest element of the poset if
• 𝑎 is the least element of the poset if
Theorems
• Every finite poset has a minimal element
• Every finite poset has a maximal element
Example:
Maximal Elements?
Minimal Elements?
Greatest Element?
Least Element?
Upper Bounds and Lower Bounds
Given a subset 𝐴 of a poset (𝑆, ≼) we say
• 𝑢 is an upper bound of 𝐴 if
• 𝑙 is a lower bound of 𝐴 if
• 𝑥 is the least upper bound of 𝐴 if
• 𝑦 is the greatest lower bound of 𝐴 if
Example:
Find the upper and lower bounds
of the subsets {c,f}, {h,i}, {c,d,e}.
Find the greatest lower bound
and the least upper bound of
{b,d} and {a,b,g}.
Topological Sorts
• Let (A,≼) be a partial ordering, and let ≤ be
a total ordering on the same set A. We say
that ≤ is a compatible total ordering for ≼
provided that for all a, b in A, whenever
a ≼ b then also a ≤ b.
• A topological sort is an algorithm which,
given a partially ordered set, generates a
compatible total ordering. In other words, it
generates the elements of A one by one in
a linear order compatible with the partial
ordering. (PERT charts)
Topological Sort Algorithm
Procedure TopologicalSort(S: finite poset)
Let q be a queue of elements of S, initially empty
While S is not empty, do begin
Choose and enqueue a minimal element s of
S onto the queue q.
Remove s from S
End
{ The queue q now contains all the elements of S,
arranged in a compatible total ordering of S. }
Topological Sort Trace