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Similar to Relations digraphs
Similar to Relations digraphs (20)
Relations digraphs
- 2. Product Sets
Definition: An ordered pair 𝑎𝑎, 𝑏𝑏 is a listing of the
objects/items 𝑎𝑎 and 𝑏𝑏 in a prescribed order: 𝑎𝑎 is the first
and 𝑏𝑏 is the second. (a sequence of length 2)
Definition: The ordered pairs 𝑎𝑎1, 𝑏𝑏1 and 𝑎𝑎2, 𝑏𝑏2 are
equal iff 𝑎𝑎1 = 𝑎𝑎2 and 𝑏𝑏1 = 𝑏𝑏2.
Definition: If 𝐴𝐴 and 𝐵𝐵 are two nonempty sets, we define
the product set or Cartesian product 𝐴𝐴 × 𝐵𝐵 as the set of
all ordered pairs 𝑎𝑎, 𝑏𝑏 with 𝑎𝑎 ∈ 𝐴𝐴 and 𝑏𝑏 ∈ 𝐵𝐵:
𝐴𝐴 × 𝐵𝐵 = 𝑎𝑎, 𝑏𝑏 𝑎𝑎 ∈ 𝐴𝐴 and 𝑏𝑏 ∈ 𝐵𝐵}
© S. Turaev, CSC 1700 Discrete Mathematics 2
- 3. Product Sets
Example: Let 𝐴𝐴 = 1,2,3 and 𝐵𝐵 = 𝑟𝑟, 𝑠𝑠 , then
𝐴𝐴 × 𝐵𝐵 =
𝐵𝐵 × 𝐴𝐴 =
© S. Turaev, CSC 1700 Discrete Mathematics 3
- 4. Product Sets
Theorem: For any two finite sets 𝐴𝐴 and 𝐵𝐵,
𝐴𝐴 × 𝐵𝐵 = 𝐴𝐴 ⋅ 𝐵𝐵 .
Proof: Use multiplication principle!
© S. Turaev, CSC 1700 Discrete Mathematics 4
- 5. Definitions:
Let 𝐴𝐴 and 𝐵𝐵 be nonempty sets. A relation 𝑅𝑅 from 𝐴𝐴
to 𝐵𝐵 is a subset of 𝐴𝐴 × 𝐵𝐵.
If 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐵𝐵 and 𝑎𝑎, 𝑏𝑏 ∈ 𝑅𝑅, we say that 𝑎𝑎 is related
to 𝑏𝑏 by 𝑅𝑅, and we write 𝑎𝑎 𝑅𝑅 𝑏𝑏.
If 𝑎𝑎 is not related to 𝑏𝑏 by 𝑅𝑅, we write 𝑎𝑎 𝑅𝑅 𝑏𝑏.
If 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐴𝐴, we say 𝑅𝑅 is a relation on 𝐴𝐴.
Relations & Digraphs
© S. Turaev, CSC 1700 Discrete Mathematics 5
- 6. Example 1: Let 𝐴𝐴 = 1,2,3 and 𝐵𝐵 = 𝑟𝑟, 𝑠𝑠 . Then
𝑅𝑅 = 1, 𝑟𝑟 , 2, 𝑠𝑠 , 3, 𝑟𝑟 ⊆ 𝐴𝐴 × 𝐵𝐵
is a relation from 𝐴𝐴 to 𝐵𝐵.
Example 2: Let 𝐴𝐴 and 𝐵𝐵 are sets of positive integer
numbers. We define the relation 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐵𝐵 by
𝑎𝑎 𝑅𝑅 𝑏𝑏 ⇔ 𝑎𝑎 = 𝑏𝑏
Relations & Digraphs
© S. Turaev, CSC 1700 Discrete Mathematics 6
- 7. Example 3: Let 𝐴𝐴 = 1,2,3,4,5 . The relation 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐴𝐴 is
defined by
𝑎𝑎 𝑅𝑅 𝑏𝑏 ⇔ 𝑎𝑎 < 𝑏𝑏
Then 𝑅𝑅 =
Example 4: Let 𝐴𝐴 = 1,2,3,4,5,6,7,8,9,10 . The relation
𝑅𝑅 ⊆ 𝐴𝐴 × 𝐴𝐴 is defined by
𝑎𝑎 𝑅𝑅 𝑏𝑏 ⇔ 𝑎𝑎|𝑏𝑏
Then 𝑅𝑅 =
Relations & Digraphs
© S. Turaev, CSC 1700 Discrete Mathematics 7
- 8. Definition: Let 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐵𝐵 be a relation from 𝐴𝐴 to 𝐵𝐵.
The domain of 𝑅𝑅, denoted by Dom 𝑅𝑅 , is the set of
elements in 𝐴𝐴 that are related to some element in
𝐵𝐵.
The range of 𝑅𝑅, denoted by Ran 𝑅𝑅 , is the set of
elements in 𝐵𝐵 that are second elements of pairs in
𝑅𝑅.
Relations & Digraphs
© S. Turaev, CSC 1700 Discrete Mathematics 8
- 9. Relations & Digraphs
Example 5: Let 𝐴𝐴 = 1,2,3 and 𝐵𝐵 = 𝑟𝑟, 𝑠𝑠 .
𝑅𝑅 = 1, 𝑟𝑟 , 2, 𝑠𝑠 , 3, 𝑟𝑟
Dom R =
Ran R =
Example 6: Let 𝐴𝐴 = 1,2,3,4,5 . The relation 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐴𝐴 is
defined by 𝑎𝑎 𝑅𝑅 𝑏𝑏 ⇔ 𝑎𝑎 < 𝑏𝑏
Dom R =
Ran R =
© S. Turaev, CSC 1700 Discrete Mathematics 9
- 10. The Matrix of a Relation
Definition: Let 𝐴𝐴 = 𝑎𝑎1, 𝑎𝑎2, … , 𝑎𝑎 𝑚𝑚 , 𝐵𝐵 = 𝑏𝑏1, 𝑏𝑏2, … , 𝑏𝑏𝑛𝑛
and 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐵𝐵 be a relation. We represent 𝑅𝑅 by the 𝑚𝑚 ×
𝑛𝑛 matrix 𝐌𝐌𝑅𝑅 = [𝑚𝑚𝑖𝑖𝑖𝑖], which is defined by
𝑚𝑚𝑖𝑖𝑖𝑖 = �
1, 𝑎𝑎𝑖𝑖, 𝑏𝑏𝑗𝑗 ∈ 𝑅𝑅
0, 𝑎𝑎𝑖𝑖, 𝑏𝑏𝑗𝑗 ∉ 𝑅𝑅
The matrix 𝐌𝐌𝑅𝑅 is called the matrix of 𝑅𝑅.
Example: Let 𝐴𝐴 = 1,2,3 and 𝐵𝐵 = 𝑟𝑟, 𝑠𝑠 .
𝑅𝑅 = 1, 𝑟𝑟 , 2, 𝑠𝑠 , 3, 𝑟𝑟 𝐌𝐌𝑅𝑅 =
© S. Turaev, CSC 1700 Discrete Mathematics 10
- 11. The Digraph of a Relation
Definition: If 𝐴𝐴 is finite and 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐴𝐴 is a relation. We
represent 𝑅𝑅 pictorially as follows:
Draw a small circle, called a vertex/node, for each
element of 𝐴𝐴 and label the circle with the
corresponding element of 𝐴𝐴.
Draw an arrow, called an edge, from vertex 𝑎𝑎𝑖𝑖 to
vertex 𝑎𝑎𝑗𝑗 iff 𝑎𝑎𝑖𝑖 𝑅𝑅 𝑎𝑎𝑗𝑗.
The resulting pictorial representation of 𝑅𝑅 is called a
directed graph or digraph of 𝑅𝑅.
© S. Turaev, CSC 1700 Discrete Mathematics 11
- 12. The Digraph of a Relation
Example: Let 𝐴𝐴 = 1, 2, 3, 4 and
𝑅𝑅 = 1,1 , 1,2 , 2,1 , 2,2 , 2,3 , 2,4 , 3,4 , 4,1
The digraph of 𝑅𝑅:
Example: Let 𝐴𝐴 = 1, 2, 3, 4 and
Find the relation 𝑅𝑅:
© S. Turaev, CSC 1700 Discrete Mathematics
1
2
3
4
12
- 13. The Digraph of a Relation
Definition: If 𝑅𝑅 is a relation on a set 𝐴𝐴 and 𝑎𝑎 ∈ 𝐴𝐴, then
the in-degree of 𝑎𝑎 is the number of 𝑏𝑏 ∈ 𝐴𝐴 such that
𝑏𝑏, 𝑎𝑎 ∈ 𝑅𝑅;
the out-degree of 𝑎𝑎 is the number of 𝑏𝑏 ∈ 𝐴𝐴 such
that 𝑎𝑎, 𝑏𝑏 ∈ 𝑅𝑅.
Example: Consider the digraph:
List in-degrees and out-degrees of all vertices.
© S. Turaev, CSC 1700 Discrete Mathematics
1
2
3
4
13
- 14. The Digraph of a Relation
Example: Let 𝐴𝐴 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑 and let 𝑅𝑅 be the relation on
𝐴𝐴 that has the matrix
𝐌𝐌𝑅𝑅 =
1 0
0 1
0 0
0 0
1 1
0 1
1 0
0 1
Construct the digraph of 𝑅𝑅 and list in-degrees and out-
degrees of all vertices.
© S. Turaev, CSC 1700 Discrete Mathematics 14
- 15. The Digraph of a Relation
Example: Let 𝐴𝐴 = 1,4,5 and let 𝑅𝑅 be given the digraph
Find 𝐌𝐌𝑅𝑅 and 𝑅𝑅.
© S. Turaev, CSC 1700 Discrete Mathematics
1 4
5
15
- 16. Paths in Relations & Digraphs
Definition: Suppose that 𝑅𝑅 is a relation on a set 𝐴𝐴.
A path of length 𝑛𝑛 in 𝑅𝑅 from 𝑎𝑎 to 𝑏𝑏 is a finite sequence
𝜋𝜋 ∶ 𝑎𝑎, 𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑛𝑛−1, 𝑏𝑏
beginning with 𝑎𝑎 and ending with 𝑏𝑏, such that
𝑎𝑎 𝑅𝑅 𝑥𝑥1, 𝑥𝑥1 𝑅𝑅 𝑥𝑥2, … , 𝑥𝑥𝑛𝑛−1 𝑅𝑅 𝑏𝑏.
Definition: A path that begins and ends at the same
vertex is called a cycle:
𝜋𝜋 ∶ 𝑎𝑎, 𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑛𝑛−1, 𝑎𝑎
© S. Turaev, CSC 1700 Discrete Mathematics 16
- 17. Paths in Relations & Digraphs
Example: Give the examples for paths of length 1,2,3,4
and 5.
© S. Turaev, CSC 1700 Discrete Mathematics
1 2
43
5
17
- 18. Paths in Relations & Digraphs
Definition: If 𝑛𝑛 is a fixed number, we define a relation 𝑅𝑅 𝑛𝑛
as follows: 𝑥𝑥 𝑅𝑅𝑛𝑛
𝑦𝑦 means that there is a path of length 𝑛𝑛
from 𝑥𝑥 to 𝑦𝑦.
Definition: We define a relation 𝑅𝑅∞
(connectivity relation
for 𝑅𝑅) on 𝐴𝐴 by letting 𝑥𝑥 𝑅𝑅∞
𝑦𝑦 mean that there is some
path from 𝑥𝑥 to 𝑦𝑦.
Example: Let 𝐴𝐴 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑, 𝑒𝑒 and
𝑅𝑅 = 𝑎𝑎, 𝑎𝑎 , 𝑎𝑎, 𝑏𝑏 , 𝑏𝑏, 𝑐𝑐 , 𝑐𝑐, 𝑒𝑒 , 𝑐𝑐, 𝑑𝑑 , 𝑑𝑑, 𝑒𝑒 .
Compute (a) 𝑅𝑅2
; (b) 𝑅𝑅3
; (c) 𝑅𝑅∞
.
© S. Turaev, CSC 1700 Discrete Mathematics 18
- 19. Paths in Relations & Digraphs
Let 𝑅𝑅 be a relation on a finite set 𝐴𝐴 = 𝑎𝑎1, 𝑎𝑎2, … , 𝑎𝑎𝑛𝑛 , and
let 𝐌𝐌𝑅𝑅 be the 𝑛𝑛 × 𝑛𝑛 matrix representing 𝑅𝑅.
Theorem 1: If 𝑅𝑅 is a relation on 𝐴𝐴 = 𝑎𝑎1, 𝑎𝑎2, … , 𝑎𝑎𝑛𝑛 , then
𝐌𝐌𝑅𝑅2 = 𝐌𝐌𝑅𝑅 ⊙ 𝐌𝐌𝑅𝑅.
Example: Let 𝐴𝐴 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑, 𝑒𝑒 and
𝑅𝑅 = 𝑎𝑎, 𝑎𝑎 , 𝑎𝑎, 𝑏𝑏 , 𝑏𝑏, 𝑐𝑐 , 𝑐𝑐, 𝑒𝑒 , 𝑐𝑐, 𝑑𝑑 , 𝑑𝑑, 𝑒𝑒 .
© S. Turaev, CSC 1700 Discrete Mathematics 19
- 20. Paths in Relations & Digraphs
Example: Let 𝐴𝐴 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑, 𝑒𝑒 and
𝑅𝑅 = 𝑎𝑎, 𝑎𝑎 , 𝑎𝑎, 𝑏𝑏 , 𝑏𝑏, 𝑐𝑐 , 𝑐𝑐, 𝑒𝑒 , 𝑐𝑐, 𝑑𝑑 , 𝑑𝑑, 𝑒𝑒 .
𝐌𝐌𝑅𝑅 =
1 1
0 0
0 0
1 0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
0
Compute 𝐌𝐌𝑅𝑅2.
© S. Turaev, CSC 1700 Discrete Mathematics 20
- 21. Reflexive & Irreflexive Relations
Definition:
A relation 𝑅𝑅 on a set 𝐴𝐴 is reflexive if 𝑎𝑎, 𝑎𝑎 ∈ 𝑅𝑅 for
all 𝑎𝑎 ∈ 𝐴𝐴, i.e., if 𝑎𝑎 𝑅𝑅 𝑎𝑎 for all 𝑎𝑎 ∈ 𝐴𝐴.
A relation 𝑅𝑅 on a set 𝐴𝐴 is irreflexive if 𝑎𝑎 𝑅𝑅 𝑎𝑎 for all
𝑎𝑎 ∈ 𝐴𝐴.
Example:
Δ = 𝑎𝑎, 𝑎𝑎 | 𝑎𝑎 ∈ 𝐴𝐴 , the relation of equality on the
set 𝐴𝐴.
𝑅𝑅 = 𝑎𝑎, 𝑏𝑏 ∈ 𝐴𝐴 × 𝐴𝐴| 𝑎𝑎 ≠ 𝑏𝑏 , the relation of
inequality on the set 𝐴𝐴.
© S. Turaev, CSC 1700 Discrete Mathematics 21
- 22. Reflexive & Irreflexive Relations
Exercise: Let 𝐴𝐴 = 1, 2, 3 , and let 𝑅𝑅 = 1,1 , 1,2 .
Is 𝑅𝑅 reflexive or irreflexive?
Exercise: How is a reflexive or irreflexive relation
identified by its matrix?
Exercise: How is a reflexive or irreflexive relation
characterized by the digraph?
© S. Turaev, CSC 1700 Discrete Mathematics 22
- 23. (A-, Anti-) Symmetric Relations
Definition:
A relation 𝑅𝑅 on a set 𝐴𝐴 is symmetric if whenever
𝑎𝑎 𝑅𝑅 𝑏𝑏, then 𝑏𝑏 𝑅𝑅 𝑎𝑎.
A relation 𝑅𝑅 on a set 𝐴𝐴 is asymmetric if whenever
𝑎𝑎 𝑅𝑅 𝑏𝑏, then 𝑏𝑏 𝑅𝑅 𝑎𝑎.
A relation 𝑅𝑅 on a set 𝐴𝐴 is antisymmetric if whenever
𝑎𝑎 𝑅𝑅 𝑏𝑏 and 𝑏𝑏 𝑅𝑅 𝑎𝑎, then 𝑎𝑎 = 𝑏𝑏.
© S. Turaev, CSC 1700 Discrete Mathematics 23
- 24. (A-, Anti-) Symmetric Relations
Example: Let 𝐴𝐴 = 1, 2, 3, 4, 5, 6 and let
𝑅𝑅 = 𝑎𝑎, 𝑏𝑏 ∈ 𝐴𝐴 × 𝐴𝐴 | 𝑎𝑎 < 𝑏𝑏
Is 𝑅𝑅 symmetric, asymmetric or antisymmetric?
Symmetry:
Asymmetry:
Antisymmetry:
© S. Turaev, CSC 1700 Discrete Mathematics 24
- 25. (A-, Anti-) Symmetric Relations
Example: Let 𝐴𝐴 = 1, 2, 3, 4 and let
𝑅𝑅 = 1,2 , 2,2 , 3,4 , 4,1
Is 𝑅𝑅 symmetric, asymmetric or antisymmetric?
Example: Let 𝐴𝐴 = ℤ+
and let
𝑅𝑅 = 𝑎𝑎, 𝑏𝑏 ∈ 𝐴𝐴 × 𝐴𝐴 | 𝑎𝑎 divides 𝑏𝑏
Is 𝑅𝑅 symmetric, asymmetric or antisymmetric?
© S. Turaev, CSC 1700 Discrete Mathematics 25
- 26. (A-, Anti-) Symmetric Relations
Exercise: How is a symmetric, asymmetric or
antisymmetric relation identified by its matrix?
Exercise: How is a symmetric, asymmetric or
antisymmetric relation characterized by the digraph?
© S. Turaev, CSC 1700 Discrete Mathematics 26
- 27. Transitive Relations
Definition: A relation 𝑅𝑅 on a set 𝐴𝐴 is transitive if
whenever 𝑎𝑎 𝑅𝑅 𝑏𝑏 and 𝑏𝑏 𝑅𝑅 𝑐𝑐 then 𝑎𝑎 𝑅𝑅 𝑐𝑐.
Example: Let 𝐴𝐴 = 1, 2, 3, 4 and let
𝑅𝑅 = 1,2 , 1,3 , 4,2
Is 𝑅𝑅 transitive?
Example: Let 𝐴𝐴 = ℤ+
and let
𝑅𝑅 = 𝑎𝑎, 𝑏𝑏 ∈ 𝐴𝐴 × 𝐴𝐴 | 𝑎𝑎 divides 𝑏𝑏
Is 𝑅𝑅 transitive?
© S. Turaev, CSC 1700 Discrete Mathematics 27
- 28. Transitive Relations
Exercise: Let 𝐴𝐴 = 1,2,3 and 𝑅𝑅 be the relation on 𝐴𝐴
whose matrix is
𝐌𝐌𝑅𝑅 =
1 1 1
0 0 1
0 0 1
Show that 𝑅𝑅 is transitive. (Hint: Check if 𝐌𝐌𝑅𝑅 ⊙
2
= 𝐌𝐌𝑅𝑅)
Exercise: How is a transitive relation identified by its
matrix?
Exercise: How is a transitive relation characterized by the
digraph?
© S. Turaev, CSC 1700 Discrete Mathematics 28
- 29. Equivalence Relations
Definition: A relation 𝑅𝑅 on a set 𝐴𝐴 is called an equi-
valence relation if it is reflexive, symmetric and transitive.
Example: Let 𝐴𝐴 = 1, 2, 3, 4 and let
𝑅𝑅 = 1,1 , 1,2 , 2,1 , 2,2 , 3,4 , 4,3 , 3,3 , 4,4 .
Then 𝑅𝑅 is an equivalence relation.
Example: Let 𝐴𝐴 = ℤ and let
𝑅𝑅 = 𝑎𝑎, 𝑏𝑏 ∈ 𝐴𝐴 × 𝐴𝐴 ∶ 𝑎𝑎 ≡ 𝑏𝑏 mod 2 .
Show that 𝑅𝑅 is an equivalence relation.
© S. Turaev, CSC 1700 Discrete Mathematics 29