MATH 213 Chapter 9: Relations

Dr. Eric Bancroft

Fall 2013

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 R6 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 + y2 = 8

1 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?

Properties of Relations

Definition 3 (Reflexive). A relation R on a set A is said to be reflexive if and only if

a R a 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.

2 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

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) ∈ R1}

Union: R1 ∪ R2 = {(a, b) | (a, b) ∈ R1 or (a, b) ∈ R1}

Diﬀerence: R1 − R2 = {(a, b) | (a, b) ∈ R1 but (a, b) ∈/ R1}

1 Symmetric Diﬀerence: R1 ⊕ R2 = {(a, b) | (a, b) ∈ R1 or (a, b) ∈ R2 but (a, b) ∈/ R1 ∩ R2}

1In the portfolio problem I used ∆ to denote symmetric diﬀerence before I realized that this book uses ⊕ instead.

3 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 = {}. Find the following:

(a) R1 ∩ R2

(b) R1 ∪ R2

(d) R2 − R1

(e) R1 ◦ R2

(f) R1 ⊕ R2

4 Deﬁnition 8. Let R be a recursion on a set A. Then we deﬁne the powers of R as

R1 = R Rn = Rn−1 ◦ R = R ◦ R ◦ · · · ◦ R, for n > 1 | {z } 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)}

n Theorem 1. A relation R on a set A is transitive if and only if R ⊆ 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.

Deﬁnition 10 (Selection Operator). Let R be an n-art relation and C a condition that elements in R may satisfy. The selection operator sC maps the n-ary relation R to the n-ary relation of all n-tuples from R that satisfy the condition C.

5 Deﬁnition 11 (Projection). The projection Pi1,i2,...,im , where i1 < i2 < ··· < im, maps the n-tuple

(a1, a2, . . . , an) to the m-tuple (ai1 , ai2 , . . . , aim ) where m < n.

Example 8. Let S = {States}, A = N, Z = {n ∈ N | 0 < n < 10000}, and let R be the relation on S × F × Z where A represents average annual snowfall and Z represents ZIP codes.

9.3 - Representing Relations

In addition to charts, sets, and graphs we can represent relations using:

1. A zero-one matrix MR = [mij], where ( 1 if (ai, bj) ∈ R mij = 0 if (ai, bj) ∈/ R

(This can be used for any relation from A to B.)

2.A digraph (or directed graph).

Deﬁnition 12 (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.

a b Example 9.

d c

Example 10. 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.

6 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] • MRn = MR

7 Example 11. 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 1 0 0 1 1 Example 12. Let MR = 0 0 1, MS = 0 0 0, MP = 0 0 1. Compute MRn , MSn , 1 1 0 1 1 1 0 0 0 MP n , to determine if the relations R, S, and P are transitive.

8 9.4 - Closures of Relations

Definition 13 (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 14 (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 13. 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 15. 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 14. Let A = {a, b, c, d} and R = {(a, b), (b, a), (a, d), (d, b), (c, c), (c, b)}.

Deﬁnition 16 (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∗ = Ri i=1

9 Example 15. Let R be the relation on the set of all people in the world that contains (a, b) iﬀ a n ∗ has met b. What is R , where n ∈ Z+? What is R ?

Deﬁnition 17 (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 16. 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 n [ R∗ = Ri i=1 and n _ [i] MR∗ = MR i=1 (It turns out that this is still not the most eﬃcient way of computing R∗.)

Example 17. R = {(1, 1), (1, 2), (2, 4), (3, 1), (4, 2)}

10 Warshall’s Algorithm

Deﬁnition 18 (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 (k)i (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 18. R = {(1, 1), (1, 2), (2, 4), (3, 1), (4, 2)}

1 2

3 4

11 9.5 - Equivalence Relations

Deﬁnition 19 (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.2 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 19.

(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.

2As 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.

12 (c) a ∼ b if and only if a ≡ b (mod 5).

(d) a ∼ b if and only if a and b say the same thing.

(e) [4]5

(f)

Deﬁnition 20 (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.).