Chapter 9: Relations
9.1 - Relations
Definition 1 (Relation). Let A and B be sets. A binary relation from A to B is a subset R ⊆ A × B, i.e., R is a set
of ordered pairs where the first element from each pair is from A and the second element is from B.
If (a, b) ∈ R then we write a R b or a ∼ b (read “a relates/is related to b [by R]”). If (a, b) ∈
/ R, then we write a 6R b or
a b.
We can represent relations graphically or with a chart in addition to a set description.
Example 1. A = {0, 1, 2}, B = {1, 2, 3}, R = {(1, 1), (2, 1), (2, 2)}
Example 2.
(a) “Parent”
(b) ∀x, y ∈ Z, x R y ⇐⇒ x2 + y 2 = 8
(c) A = {0, 1, 2}, B = {1, 2, 3}, a R b ⇐⇒ a + b ≥ 3
(d)
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Note: All functions are relations, but not all relations are functions.
Definition 2. If A is a set, then a relation on A is a relation from A to A.
Example 3. How many relations are there on a set with. . .
(a) two elements?
(b) n elements?
(c) 14 elements?
Properties of Relations
Definition 3 (Reflexive). A relation R on a set A is said to be reflexive if and only if
aRa
for all a ∈ A.
Definition 4 (Symmetric). A relation R on a set A is said to be symmetric if and only if
a R b =⇒ b R a
for all a, b ∈ A.
Definition 5 (Anitsymmetric). A relation R on a set A is said to be antisymmetric if and only if
a R b and b R a =⇒ a = b
for all a, b ∈ A.
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Definition 6 (Transitive). A relation R on a set A is said to be transitive if and only if
a R b and b R c =⇒ a R c
for all a, b, c ∈ A.
Example 4. Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, or
transitive, where (a, b) ∈ R if and only if
(a) a 6= b
(b) a ≥ b2
Example 5. Determine whether the relation R on the set of all web pages is reflexive, symmetric, antisymmetric, or
transitive, where (a, b) ∈ R if and only if
(a) Everyone who has visited page a has also visited page b.
(b) There are no common links found on both page a and page b
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Combining Relations
We can combine two relations R1 and R2 from A to B using the following operations (these all come from set operations
that we’ve discussed previously(:
Intersection: R1 ∩ R2 = {(a, b) | (a, b) ∈ R1 and (a, b) ∈ R2 }
Union: R1 ∪ R2 = {(a, b) | (a, b) ∈ R1 or (a, b) ∈ R2 }
Difference: R1 − R2 = {(a, b) | (a, b) ∈ R1 but (a, b) ∈
/ R2 }
Symmetric Difference: R1 ⊕ R2 = {(a, b) | (a, b) ∈ R1 or (a, b) ∈ R2 but (a, b) ∈
/ R1 ∩ R2 }
Definition 7 (Composition). If R1 is a relation from A to B and R2 is a relation from B to C, then we define
R2 ◦ R1 = {(a, c) | a ∈ A, c ∈ C, and there exists b ∈ B such that (a, b) ∈ R1 and (b, c) ∈ R2 }
Example 6. Let R1 and R2 be relations on Z where R1 = {(a, b) | a|b} and R2 = {(a, b) | a is a multiple of b}. Find
the following:
(a) R1 ∩ R2
(b) R1 ∪ R2
(c) R1 − R2
(d) R2 − R1
(e) R1 ⊕ R2
(f) R1 ◦ R2
(g) R2 ◦ R1
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Definition 8. Let R be a relation on a set A. Then we define the powers of R as
R1 = R
Rn = Rn−1 ◦ R = R
◦ · · · ◦ R},
| ◦ R {z
for n > 1
n copies of R
Example 7. A = {1, 2, 3, 4, 5}, R = {(1, 1), (1, 2), (1, 3), (2, 3), (2, 4), (3, 1), (3, 4), (3, 5)}
Theorem 1. A relation R on a set A is transitive if and only if Rn ⊆ R for all n ∈ Z+ .
9.2 - n-ary Relations
Definition 9 (n-ary Relation). Let A1 , A2 , . . . , An be sets. An n-ary relation on these sets is a subset
R ⊆ A1 × A2 × · · · × An .
The sets A1 , A2 , . . . , An are called the domains of the relation and n is the degree.
9.3 - Representing Relations
In addition to charts, sets, and graphs we can represent relations using:
1. A zero-one matrix MR = [mij ], where
(
mij =
if (ai , bj ) ∈ R
if (ai , bj ) ∈
/R
1
0
(This can be used for any relation from A to B.)
2. A digraph (or directed graph).
Definition 10 (Digraph). A digraph consists of a set V of vertices or nodes together with a set E of ordered pairs of
elements of V called edges or arcs. Given an edge (a, b), a is called the initial vertex of the edge and b is called the
terminal vertex of the edge. An edge from a vertex to itself, (a, a), is call a loop.
Example 8.
a
b
d
c
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Example 9. Draw the “divides” relation on the set {2, 3, 4, 5, 6, 7, 8, 9} both as a digraph and as a 0-1 matrix.
We can use digraphs and 0-1 matrices to identify whether or not a relation is reflexive, symmetric, antisymmetric, or
transitive.
Reflexive:
Symmetric:
Antisymmetric:
Transitive:
Theorem 2. Given relations R1 , R2 , and R on a set A with matrix representations MR1 , MR2 , and MR , respectively,
then
• MR1 ∪R2 = MR1 ∨ MR2
• MR1 ∩R2 = MR1 ∧ MR2
• MR1 ◦R2 = MR2 MR1
[n]
• MR n = MR
6
Example 10. Let R1 = {(1, 2), (2, 1), (2, 2), (3, 3)} and R2 = {(1, 1), (1, 2), (1, 3), (3, 2)} be binary relations on the
set A = {1, 2, 3}. Find MR1 and MR2 and then use them to find MR1 ∪R2 , MR1 ∩R2 , and MR1 ◦R2 . Verify by computing
R1 ∪ R2 , R1 ∩ R2 , and R1 ◦ R2 without matrices.



1 0 1
1
Example 11. Let MR = 0 0 1, MS = 0
1 1 0
1
if the relations R, S, and P are transitive.
1
0
1


0
0
0, MP = 0
1
0
7
1
0
0

1
1. Compute MRn , MS n , MP n , to determine
0
9.4 - Closures of Relations
Definition 11 (Reflexive Closure). Let R be a relation on a set A. The reflexive closure of R is the smallest relation
containing R that is reflexive. We denote the reflexive closure by R ∪ ∆ where ∆ = {(a, a) | a ∈ A}.
Definition 12 (Symmetric Closure). Let R be a relation on a set A. The symmetric closure of R is the smallest
relation containing R that is symmetric. We denote the reflexive closure by R ∪ R−1 where R−1 = {(b, a) | (a, b) ∈ R}.
Example 12. Let R = {(1, 1), (1, 2), (2, 4), (3, 1), (4, 2)} be a relation on the set A = {1, 2, 3, 4}. Find the reflexive
closure and symmetric closure of R.
Definition 13. A path from a to b in a digraph G is a sequence of edges (x0 , x1 ), (x1 , x2 ), (x2 , x3 ),. . . , (xn−1 , xn )
where n ∈ N, a = x0 , and b = xn . This path is denoted by x0 , x1 , . . . , xn and has length n. An empty set of edges is
viewed as a path of length 0 from a to a. A path of length n ≥ 1 that begins and ends at the same vertex is called a
circuit or cycle.
Example 13. Let A = {a, b, c, d} and R = {(a, b), (b, a), (a, d), (d, b), (c, c), (c, b)}.
Definition 14 (The Connectivity Relation). Let R be a relation on the set A. The connectivity relation R∗ consists of
all pairs (a, b) such that there is path of length at least 1 from a to b in R. Alternatively,
R∗ =
∞
[
i=1
8
Ri
Example 14. Let R be the relation on the set of all people in the world that contains (a, b) iff a has met b. What is
Rn , where n ∈ Z+ ? What is R∗ ?
Definition 15 (Transitive Closure). Let R be a relation on a set A. The transitive closure of R is the smallest relation
containing R that is transitive.
Theorem 3. The transitive closure of R is R∗ .
Example 15. R = {(1, 1), (1, 2), (2, 4), (3, 1), (4, 2)}
It turns out that if A is a set with n elements and R is a relation on A, then any time there is a path of length 1 or
more from a to b in R there there is a path of length n or less from a to b in R. This means that
R∗ =
n
[
Ri
i=1
and
MR ∗ =
n
_
[i]
MR
i=1
(It turns out that this is still not the most efficient way of computing R∗ .)
Example 16. R = {(1, 1), (1, 2), (2, 4), (3, 1), (4, 2)}
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Warshall’s Algorithm
Definition 16 (Interior Vertex). An interior vertex is any vertex in a path that is not the initial vertex or terminal
vertex. (The initial or terminal vertex could be an interior vertex as long as the path visits it again without starting or
ending there.)
h
i
(k)
(k)
Theorem 4 (Warshall’s Algorithm). Wk = wij , where wij = 1 if and only if there is a path from vi to vj such
that all interior vertices are in the set {v1 , . . . , vk }. (Note: the first and last vertices in the path can be outside, length
1 paths always count since there are no interior vertices (vacuously true), and the path doesn’t have to visit all of the
vertices v1 , . . . , vk .)
Example 17. R = {(1, 1), (1, 2), (2, 4), (3, 1), (4, 2)}
1
2
3
4
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9.5 - Equivalence Relations
Definition 17 (Equivalence Relation). A relation R on a set A is called an equivalence relation if and only if R is
• reflexive,
• symmetric,
• and transitive.
If two elements a and b are related by an equivalence relation, then we write a ∼ b.1 All elements that are related to
an element a ∈ A form the equivalence class of a, denoted by [a]R (or simply [a] if there is only one relation under
consideration):
[a]R = {b | (b, a) ∈ R}
a is called a representative of the equivalence class [a]R .
Theorem 5. Given an equivalence relation R on a set A and given two elements a, b ∈ A, the following are equivalent:
1. a ∼ b
2. [a]R = [b]R
3. [a]R ∩ [b]R = ∅
Example 18.
(a) a ∼ b if and only if a and b have the same gender.
(b) a ∼ b if and only if a and b have the same first name.
1 As was mentioned earlier in the notes, some books use the ∼ notation to denote any relation, but in this book it is only used if element
are related under an equivalence relation.
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(c) a ∼ b if and only if a ≡ b (mod 5).
(d) [4]5
(e) a ∼ b if and only if a and b say the same thing.
(f)
Definition 18 (Partition). A partition of a set S is a collection Π = {A1 , A2 , . . . , Am } of nonempty, pairwise disjoint
subsets of S such that every element of S is in one of the subsets Ai . We say that Π partitions S (or “Π is a partition
of S”).
Theorem 6. Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S.
Conversely, any partition Π of S defines an equivalence relation on S whose equivalence classes are the sets in Π.
Equivalence relations give us a way to identify some notion of “sameness” (same name, same remainder, same set of a
partition, etc.).
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