relations and diagraphs

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					                               Module ECSC301
               Elements of Discrete and Continuous Mathematics

                                      Lecture 6
                                Relations and Digraphs

Product Sets

• Ordered Pair
  o An ordered pair (a, b) is a listing of the objects a and b in a prescribed order,
    with ‘a’ appearing first and ‘b’ appearing second
  o Generally, (a, b) is not equal to (b, a)

• Product Set or Cartesian Product
  o We define the Product Set or Cartesian Product A x B as the set of all ordered
    pairs (a, b) with a ∈ A and b ∈ B. Thus
          A x B = {(a, b) | a ∈ A and b ∈ B}

Theorem 1       For any two finite, non–empty sets A and B, |A x B| = |A| x |B|

• Partition or Quotient Set
  o A Partition or Quotient Set of a non–empty set A is a collection P of non–empty
    subsets of A such that
    1. Each element of A belongs to one of the sets in P (i.e. P = A)
    2. If A1 and A2 are distinct elements of P (i.e. A1 ∈ P, A2 ∈ P, A1 ≠ A2),
       then A1 ∩ A2 = ∅. The sets in P are called the blocks or cells of the partition

Relations and Diagraphs

• Relation
  o Relations are common in Mathematics
  o In propositional logic, we saw that one proposition could be equivalent to
    another, e.g. p ↔ q is equivalent to (p → q) (q → p)
  o In everyday life, we have all sorts of relation, e.g. John is studying Biology,
    Alice is studying Computer Science, etc.
  o So we have a set A = {John, Alice, Bob, ...} and another set B = {Biology,
    Computer Science, Geography, ...}
  o In set terminology, we often refer to the relation of a set as being a subset of
    another set
  o For our purposes, we will only be studying a binary relation, i.e. the relation
    between one element of a set to an element of the same or different set
  o Let A and B are non–empty sets. A relation R from A to B is subset of A x B
  o If R ⊆ A x B and (a, b) ∈ R, we say that a is related to b by R which is written
    as aRb
  o R is a statement about ordered pairs (a, b) belonging to A x B, i.e. it will tell us
    whether a is related to b or a is not related to b
  o Such statement is either true or false
  o Frequently, A and B are equal. In this case, we often say that R ⊆ A x A is a
    relation on A instead of a relation from A to A
  o In such cases, we often say that R ⊆ A x A is a relation on A instead of a
    relation from A to A and we write it as aRa

• Domain and Range of Relations
  o Let R ⊆ A x B be a relation from A to B. We now define various important and
    useful sets related to R
  o The domain of R, denoted by Dom(R), is the set of elements in A which are
    related to some element in B. In other words, Dom(R), a subset of A, is the set
    of all first elements in the pairs that make up R
  o We define the range of R, denoted by Ran(R), to be the set of elements in B
    which are second elements of pairs in R, that is, all elements in B which are
    related to some element in A

Theorem 2        Let
                        R be a relation from A to B
                        A1 and A2 be subsets of A
                 Then
                        If A1⊆ A2, then R(A1) ⊆ R(A2)
                        R(A1 ∪ A2) = R(A1) ∪ R(A2)
                        R(A1 ∩ A2) ⊆ R(A1) ∩ R(A2)

• Matrix of a Relation
  o If A = {a1, a2, …, am} and B = {b1, b2,…, bn} are finite sets, containing m and n
    elements respectively and R is a relation from A to B, we can represent R by
    the m x n matrix MR = [mij], which is defined as follows:
    o mij = 1 if (ai, bj) ∈ R
    o mij = 0 if (ai, bj) ∉ R
  o The matrix MR is called the matrix of R

• Diagraphs
  o If A is a finite set and R is a relation on A, we can represent R pictorially as
    follows (see fig. 6)
    o Draw a small circle for each element of A and label the circle with the
        corresponding element of A
    o These circles are called vertices
    o Draw a directed line, called an edge, from vertex ai to vertex aj if and only if
        aiRaj (aiRaj denotes (ai, aj) ∈ R). The resulting pictorial representation of R is
        called a directed graph or diagraph of R




                               Fig. 6 Drawing a Digraph
• Paths in Relations and Digraphs
  o Suppose that R is a relation on a set A
  o A path of length n in R from a to b is a finite sequence π:a, x1, x2,…, xn-1, b,
    beginning with a and ending with b, such that aRx1, x1Rx2,…, xn-1Rb
                              b
                                                  xn-1




                                                              x3
                          a


                                             x1


                                                             x2

                                  Fig. 6 Path from a to b

Properties of Relations

• Reflexive Relation
  o A relation R on a set A is reflexive if (a, a) ∈ R for every a ∈ A (i.e. if aRa for all
    a ∈ A), i.e. a reflexive relation on a set is one for which every element is
    related to itself, see fig. 7
  o The “less than or equal to” relation is a reflexive relation defined on a set of
    integers
  o The relation “ the product of x and y is even” is reflexive on a set of even
    numbers




        Fig. 7 Reflexive Relation                 Fig. 8 Ir–reflexive Relation

• Ir–reflexive Relation
  o An ir–reflexive relation is opposite to a reflexive relation
  o In an ir–reflexive relation, no element is related to itself
                                                /
  o A relation R on a set A is ir–reflexive if aRa for every a ∈ A (i.e. for all a ∈ A,
     (a, a) ∉ R), see fig. 8
  o An example of an ir–reflexive relation is x < y
• Symmetric Relation
  o A relation R on a set A is symmetric if whenever aRb, then bRa, i.e. the relation
    R is symmetric when aRb if and only if bRa (remember that aRb means a is
    relation to b by R where R ⊆ A x B), see fig. 9
  o Thus, whenever a is related to b, b is related to a
  o It then follows that R is not symmetric if we have some a and b ∈ A with aRb,
    but bRa/




       Fig. 9 Symmetric Relation             Fig. 10 Asymmetric Relation

• Asymmetric Relation
                                                                 /
  o A relation R on a set A is asymmetric if whenever aRb, then bRa. It then
    follows that R is not asymmetric if we have some a and b ∈ A with both aRb
           /
    and bRa, see fig. 10

• Anti–symmetric Relation
  o A relation R on a set A is anti–symmetric if whenever aRb and bRa, then a = b
                                                        /
  o R is anti–symmetric if whenever a ≠ b, we have aRb but bRa, or aRb but bRa  /
  o It follows that R is not anti–symmetric if we have a and b in A and both aRb
    and bRa

• Transitive Relation
  o A relation R on a set A is transitive if whenever aRb and bRc, then aRc, i.e.
    whenever a is related to b and b is related to c, then a is related to c
  o A relation R on A is not transitive if there exist a, b and c in A so that aRb and
                /
    bRc, but aRc. If such a, b and c do not exist, then R is transitive




                             Fig. 11 Transitive Relation

Theorem 3       A relation R is transitive if and only if it satisfies the following
                property: If there is a path of length greater than 1 from vertex a to
                vertex b, there is a path of length 1 from a to b (i.e. a is related to b).
                Algebraically stated, R is transitive if and only if Rn ⊆ R for all n ≥ 1
An Example
The relation R defined over {a, b, c} by the digraph shown in fig. 12 is not reflexive
(e.g. (a, a ) ∉ R), not ir–reflexive (e.g. (c, c ) ∈ R), not symmetric (e.g. (a, c ) ∈ R,
but (c, a ) ∉ R), not anti-symmetric (e.g. (a, b) ∈ R and (b, a) ∈ R, but a ≠ b), not
transitive (e.g. (a, b) ∈ R and (b, a) ∈ R, but (a, a ) ∉ R).


                                        a




                            b                      c




                           Fig. 12 An Example of Relations

				
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Description: Relations and diagraphs basics in network security