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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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Relations and diagraphs basics in network security

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