Book: Discrete Mathematics & It's Applications Writer: Kenneth H Rosen

1. RELATIONS
TAHSIN AZIZ
COMPUTER SCIENCE AND ENGINEERING
AHSANULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY

2. REFERENCE BOOK
DISCRETE MATHEMATICS AND ITS APPLICATIONS
BY KENNETH H. ROSEN
Tahsin Aziz 2

3. RELATIONS
• Relationships between elements of sets are represented using the structure called a
relation, which is just a subset of the Cartesian product of the sets.
• Relations can be used to solve problems such as determining which pairs of cities are
linked by airline flights in a network, finding a viable order for the different phases of a
complicated project, or producing a useful way to store information in computer
databases.
• Two types
• Binary Relation
• N-ary Relation
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4. BINARY RELATION
• Let A and B be two sets. A binary relation from A to B is a subset of A × B
• A binary relation from A to B is a set R of ordered pairs where the first element of each
ordered pair comes from A and the second element comes from B.
• We use the notation aRb to denote that (a, b) ∈ R and aRb to denote that (a, b) ∉ R.
• Moreover, when (a, b) belongs to R, a is said to be related to b by R.
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5. BINARY RELATION
• Example
• Let A = {0, 1, 2} and B = {a, b}.
• Then {(0, a), (0, b), (1, a), (2, b)} is a relation from A to B.
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6. RELATIONS
• Two types
• Binary Relation √
• N-ary Relation
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7. N-ARY RELATION
• Let there are n number of sets such that A1, A2,…….,An.
• An N-ary relation on these sets is a subset of A1 × A2 ×…… × An.
• The sets A1, A2,……,An are called the domains of the relation and n is called its degree.
• It is used in database applications.
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8. RELATIONS
• Two types
• Binary Relation √
• N-ary Relation √
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9. RELATIONS
• If R = ∅ , it is called an empty relation.
• If R = A × B, it is called an universal relation.
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10. REPRESENTATIONS OF BINARY RELATIONS
• Relation can be represented in four ways.
1. Set Representation
2. Arrow Diagram
3. Tabular Matrix
4. Co-ordinate/Graph
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11. SET REPRESENTATION
• Let there are two sets A and B. Set A = {1, 2, 3, 4} and B = {x, y, z}. R is the relation from
set A to B.
• The set representation of R is:
R = { (1, y), (1, z), (3, y), (3, z), (4, x), (4, z) }
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12. REPRESENTATIONS OF BINARY RELATIONS
• Relation can be represented in four ways.
1. Set Representation √
2. Arrow Diagram
3. Tabular Matrix
4. Co-ordinate/Graph
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13. ARROW DIAGRAM
• Let there are two sets A and B. A = {1, 2, 3, 4} and B = {x, y, z}. R is the relation from set A to B.
• R = { (1, y), (1, z), (3, y), (3, z), (4, x), (4, z) }.
• The arrow diagram of R is:
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1
2
3
4
z
y
x

14. REPRESENTATIONS OF BINARY RELATIONS
• Relation can be represented in four ways.
1. Set Representation √
2. Arrow Diagram √
3. Tabular Matrix
4. Co-ordinate/Graph
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15. TABULAR MATRIX
• Let there are two sets A and B. A = {1, 2, 3, 4} and B = {x, y, z}. R is the relation from set A to B.
• R = { (1, y), (1, z), (3, y), (3, z), (4, x), (4, z) }.
• The tabular matrix form of R is:
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x y z
1 0 1 1
2 0 0 0
3 0 1 1
4 1 0 1

16. REPRESENTATIONS OF BINARY RELATIONS
• Relation can be represented in four ways.
1. Set Representation √
2. Arrow Diagram √
3. Tabular Matrix √
4. Co-ordinate/Graph
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17. CO-ORDINATE/GRAPH
• Let there are two sets A and B. A = {1, 2, 3, 4} and B = {x, y, z}. R is the relation from set A to B.
• R = { (1, y), (1, z), (3, y), (3, z), (4, x), (4, z) }.
• The co - ordinate or graph form of R is:
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z x x x
y x x
x x
1 2 3 4

18. REPRESENTATIONS OF BINARY RELATIONS
• Relation can be represented in four ways.
1. Set Representation √
2. Arrow Diagram √
3. Tabular Matrix √
4. Co-ordinate/Graph √
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19. RELATIONS AND FUNCTIONS
• Relations represent one-to-many relationships between the elements of the sets A and B,
where an element of A may be related to more than one element of B.
• A function on sets A, B represents a relation where exactly one element of B is related to
each element of A.
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20. RELATIONS AND FUNCTIONS
• Relations represent one-to-many relationships between the elements of the sets A and B,
where an element of A may be related to more than one element of B.
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0
1
2
b
a

21. RELATIONS AND FUNCTIONS
• A function on sets A, B represents a relation where exactly one element of B is related to
each element of A.
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0
1
2
b
a

22. RELATIONS ON A SET
• A relation on a set A is a relation from A to A.
• In other words, a relation on a set A is a subset of A × A
• Example
• Let A be the set {1, 2, 3, 4}, which ordered pairs are in the relation R = {(a, b) | a divides b}?
• Solution
• R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}.
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23. PROPERTIES OF BINARY RELATION
• Reflexivity
• Symmetry
• Transitivity
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24. REFLEXIVITY
• Reflexive
• A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.
• Using quantifiers it can be written that, the relation R on the set A is reflexive if ∀a a, a ∈
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25. REFLEXIVITY
• The following relations on {1, 2, 3, 4} is reflexive.
• R1 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}
• R2 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)},
• Tabular matrix representation of reflexive relation:
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1 2 3
1 1 1/0 1/0
2 1/0 1 1/0
3 1/0 1/0 1

26. REFLEXIVITY
• Irreflexive
• A relation R on a set A is called irreflexive if (a, a) ∉ R for every element of a ∈ A.
• Using quantifiers it can be written that, the relation R on the set A is irreflexive if ∀a a, a ∉
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27. REFLEXIVITY
• The following relations on {1, 2, 3, 4} are irreflexive.
• R3 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
• R4 = {(3, 4)}
• Tabular matrix representation of irreflexive relation:
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1 2 3
1 0 1/0 1/0
2 1/0 0 1/0
3 1/0 1/0 0

28. REFLEXIVITY
• Not Reflexive
• A relation R on a set A is called not reflexive if (a, a) ∉ R for some or at least one element of a
∈ A.
• Using quantifiers it can be written that, the relation R on the set A is not reflexive if ∃a a, a ∈
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29. REFLEXIVITY
• The following relations on {1, 2, 3, 4} are not reflexive.
• R5 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
• R6 = {(1, 1), (1, 2), (2, 1)}
• Tabular matrix representation of not reflexive relation:
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1 2 3
1 0 1/0 1/0
2 1/0 1 1/0
3 1/0 1/0 1

30. PROPERTIES OF BINARY RELATION
• Reflexivity √
• Symmetry
• Transitivity
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31. SYMMETRY
• Symmetric
• A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a, b) ∈ R, for all a, b ∈ A.
• Whenever aRb, then there should be bRa for every a, b ∈ A.
• Using quantifiers, it can be written as the relation R on the set A is symmetric if ∀a∀b a, b ∈
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32. SYMMETRY
• Antisymmetric
• A relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b is called
antisymmetric.
Or
• If aRb with a ≠ b exists, then bRa must not hold.
• Similarly, the relation R on the set A is antisymmetric if ∀a∀b a, b ∈ R ∧ b, a ∈ R → a = b .
• Some examples of antisymmetric relation can be:
• <, ≤ on Z (Set of all integers)
• ⊂ on P(Z) (Power set of Z)
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33. SYMMETRY
• The following relations on {1, 2, 3, 4} are antisymmetric.
• R3 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
• R4 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
• R5 = {(3, 4)}.
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34. SYMMETRY
• Not symmetric
• A relation R on a set A is called not symmetric if it is not both symmetric and antisymmetric.
• The following Relation R6 on {1, 2, 3, 4} is not symmetric as it is not both symmetric and
antisymmetric.
• R6 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
• R6 is not symmetric because for (3, 4) and (4, 1), (4, 3) and (1, 4) not in R respectively.
• R6 is not antisymmetric because for (1, 2), (2, 1) holds.
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35. SYMMETRY
• Both symmetric and antisymmetric
• A relation R on a set A can be both symmetric and antisymmetric.
• The following Relation R7 on {1, 2, 3, 4} is both symmetric and antisymmetric.
• R7 = {(1, 1), (3, 3)}
• R7 is symmetric because for all (a, b), (b, a) holds.
• R7 is antisymmetric because for there is no (a, b) and (b, a) where a ≠ b.
• So a relation is not symmetric does not mean it is antisymmetric.
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36. PROPERTIES OF BINARY RELATION
• Reflexivity √
• Symmetry √
• Transitivity
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37. TRANSITIVITY
• Transitive
• A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, for all
a, b, c ∈ A.
• Using quantifiers it can be written as the relation R on a set A is transitive if we have
∀a∀b∀c a, b ∈ R ∧ b, c ∈ R → a, c ∈ R .
• That is whenever aRb and bRc holds, then there must be aRc.
• The following relations on {1, 2, 3, 4} are transitive.
• R1 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
• R2 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
• R3 = {(3, 4)}
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38. PROPERTIES OF BINARY RELATION
• Reflexivity √
• Symmetry √
• Transitivity √
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39. COMBINING RELATIONS
• Relations from A to B are subsets of A × B.
• Two relations from A to B can be combined in any way two sets can be combined.
• So if there are two relations R1 and R2 then the following operations can be done on the sets:
• R1 ⋃ R2
• R1 ⋂ R2
• R1 − R2
• R2 − R1
• R1 ⊕ R2
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40. COMBINING RELATIONS
• Let R be a relation from a set A to a set B and S be a relation from set B to a set C.
• The composite of R and S is the relation consisting of ordered pairs (a, c), where a ∈ A, c ∈ C, and for
which there exists an element b ∈ B such that (a, b) ∈ R and (b, c) ∈ S. We denote the composite of R and
S by S ∘ R.
• Example:
• Let R is the relation from {1, 2, 3} to {1, 2, 3, 4} with R = {(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)} and S is the relation from
{1, 2, 3, 4} to {0, 1, 2} with S = {(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)}.
• The composite of the relations R and S, S ∘ R is constructed using all ordered pairs in R and ordered pairs in S,
where the second element of the ordered pair in R agrees with the first element of the ordered pair in S.
• So, S ∘ R = {(1, 0), (1, 1), (2, 1), (2, 2), (3, 0), (3, 1)}.
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41. REPRESENTING RELATIONS
• A relation can be represent by two ways:
• By using Matrix
• By using Digraph
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42. MATRIX REPRESENTATION
• A relation between finite sets can be represented using a zero–one matrix.
• Suppose that R is a relation from A = {a1, a2,……,am} to B = {b1, b2,……,bn}.
• The elements of the sets A and B have been listed in a particular, but arbitrary order. Furthermore,
when A = B we use the same ordering for A and B.
• The relation R can be represented by the matrix MR = mij , where
• mij =
1 if ai, bj ∈ R,
0 if ai, bj ∉ R.
• In other words, the zero–one matrix representing R has a 1 as its (i, j) entry when ai is related to bj , and
a 0 in this position if ai is not related to bj .
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43. MATRIX REPRESENTATION
• Suppose that A = {1, 2, 3} and B = {1, 2}. Let R be the relation from A to B containing (a,
b) if a ∈ A, b ∈ B, and a > b. What is the matrix representing R if a1 = 1, a2 = 2, and a3 =
3, and b1 = 1 and b2 = 2?
• Solution:
• R = {(2, 1), (3, 1), (3, 2)}. It can be represented as
Tahsin Aziz 43
Set B
1 2
SetA
1 0 0
2 1 0
3 1 1

44. MATRIX REPRESENTATION
• So the Matrix for R is MR =
0 0
1 0
1 1
.
• Here, 1s in MR show that the pairs (2, 1), (3, 1), and (3, 2) belong to R. The 0s show that
no other pairs belong to R.
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45. MATRIX REPRESENTATION
• The square matrix of a relation on a set can be used to determine whether the relation
has certain properties.
• Reflexivity
• Symmetry
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46. MATRIX REPRESENTATION
• Reflexive
• A relation R on A is reflexive if(a, a) ∈ R whenever a ∈ A.
• So it can be said, R is reflexive if and only if ai, ai ∈ R for i = 1, 2,……,n.
• R is reflexive if and only if mii = 1, for i = 1, 2,…….,n.
• R is reflexive if all the elements on the main diagonal of MR are equal to 1.
1 0/1 0/1
0/1 1 0/1
0/1 0/1 1
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47. MATRIX REPRESENTATION
• Symmetric
• The relation R is symmetric if (a, b) ∈ R implies that (b, a) ∈ R.
• The relation R on the set A = {a1, a2,...,an} is symmetric if and only if (aj , ai) ∈ R whenever (ai,
aj ) ∈ R.
• R is symmetric if and only if mji = 1 whenever mij = 1. and mji = 0 whenever mij = 0.
• So, R is symmetric if and only if mij = mji, for all pairs of integers i and j with i = 1, 2,……,n
and j = 1, 2,…….,n.
• According to the definition of Transpose Matrix it can be conclude, R is symmetric if and only
if MR = (MR)t
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48. MATRIX REPRESENTATION
• Antisymmetric
• The relation R is antisymmetric if and only if (a, b) ∈ R and (b, a) ∈ R imply that a = b.
• The matrix of an antisymmetric relation has the property that if mij = 1 with i ≠ j , then mji =
0.
• That means either mij = 0 or mji = 0 when i ≠ j .
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49. MATRIX REPRESENTATION
• Relation R on a set is represented by the following matrix. Determine whether R is
reflexive, symmetric, and/or antisymmetric?
MR =
1 1 0
1 1 1
0 1 1
.
• All the diagonal elements of this matrix are equal to 1, so R is reflexive.
• MR is symmetric, so R is symmetric.
• R is not antisymmetric.
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50. MATRIX REPRESENTATION
• The boolean operations AND, OR can be used to find the matrices representing the union and the
intersection of two relations.
• If R1 and R2 are relations on a set A represented by the matrices MR1
and MR2
, respectively.
• The matrix representing the union of these relations has a 1 in the positions where either MR1
or MR2
has
a 1.
• The matrix representing the intersection of these relations has a 1 in the positions where both MR1
and
MR2
have a 1.
• The matrices representing the union and intersection of these relations are
• M R1 ⋃ R2
= MR1
⋁ MR2
• M R1 ⋂ R2
= MR1
∧ MR2
.
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51. MATRIX REPRESENTATION
• The matrix for the composite of relations can be found using the boolean product of the
matrices for these relations.
• Let R is a relation from A to B and S is a relation from B to C. Suppose that A, B, and C have
m, n, and p elements, respectively. Let the zero– one matrices for S ∘ R, R, and S be MS∘R = [tij
], MR = [rij ], and MS = [sij ], respectively. These matrices have sizes m × p, m × n, and n × p
respectively.
• The ordered pair (ai, cj ) belongs to S ∘ R if and only if there is an element bk such that (ai, bk)
belongs to R and (bk, cj ) belongs to S. It follows that tij = 1 if and only if rik = skj = 1 for some
k.
• MS∘R = MR ⨀ MS.
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52. REPRESENTING RELATIONS
• A relation can be represent by two ways:
• By using Matrix √
• By using Digraph
Tahsin Aziz 52

53. DIGRAPH REPRESENTATION
• Presenting a relation using a pictorial representation is known as Digraph representation.
• Each element of the set is represented by a point, and each ordered pair is represented using
an arc with its direction indicated by an arrow.
• A directed graph, or 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).
• The vertex a is called the initial vertex of the edge (a, b), and the vertex b is called the
terminal vertex of this edge.
• An edge of the form (a, a) is represented using an arc from the vertex a back to itself which is
known as loop.
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54. DIGRAPH REPRESENTATION
• The relation R on a set A is represented by the directed graph that has the elements of A as
its vertices and the ordered pairs (a, b), where (a, b) ∈ R, as edges.
• This assignment sets up a one-to-one correspondence between the relations on a set A and
the directed graphs with A as their set of vertices.
• Every statement about relations corresponds to a statement about directed graphs, and vice
versa.
• Directed graphs give a visual display of information about relations.
• The relations from a set A to a set B can be represented by a directed graph where there is a
vertex for each element of A and a vertex for each element of B.
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55. DIGRAPH REPRESENTATION
• Directed graph can be used to determine properties of relations.
• Reflexive
• A relation is reflexive if and only if there is a loop at every vertex of the directed graph, so
that every ordered pair of the form (x, x) occurs in the relation.
• Symmetric
• A relation is symmetric if and only if for every edge between distinct vertices in its digraph
there is an edge in the opposite direction, so that (y, x) is in the relation whenever (x, y) is in
the relation.
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56. DIGRAPH REPRESENTATION
• Antisymmetric
• A relation is antisymmetric if and only if there are never two edges in opposite directions
between distinct vertices.
• Transitive
• Finally, a relation is transitive if and only if whenever there is an edge from a vertex x to a
vertex y and an edge from a vertex y to a vertex z, there is an edge from x to z. Which means
completing a triangle where each side is a directed edge with the correct direction.
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57. DIGRAPH REPRESENTATION
• The directed graph or digraph with vertices a, b, c, d and some edges.
• Edges (a, b), (a, d), (b, b), (b, d), (c, a), (c, b), and (d, b) will be look like
a
c d
b
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58. REPRESENTING RELATIONS
• A relation can be represent by two ways:
• By using Matrix √
• By using Digraph √
Tahsin Aziz 58

59. CLOSURES OF RELATIONS
• Let R be a relation on a set A.
• R may or may not have some property P, such as reflexivity, symmetry, or transitivity.
• If there is a relation S with property P containing R such that S is a subset of every
relation with property P containing R, then S is called the closure of R with respect to P.
• There are three types of closures:
• Reflexive Closure
• Symmetric Closure
• Transitive Closure
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60. CLOSURES OF RELATIONS
• Reflexive Closure
• Let R is a relation on a set A
• The reflexive closure of R can be formed by adding to R all pairs of the form (a, a) with a ∈ A,
not already in R.
• The addition of these pairs produces a new relation that is reflexive, contains R, and is
contained within any reflexive relation containing R.
• The reflexive closure of R equals R ∪ ∆ , where = ∆ {(a, a) | a ∈ A} is the diagonal relation on
A.
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61. CLOSURES OF RELATIONS
• Example:
• The relation R = {(1, 1), (1, 2), (2, 1), (3, 2)} on the set A = {1, 2, 3} is not reflexive. This can be
make reflexive by adding (2, 2) and (3, 3) to R, because these are the only pairs of the form (a,
a) that are not in R. Clearly, this new relation contains R.
• Furthermore, any reflexive relation that contains R must also contain (2, 2) and (3, 3). Because
this relation contains R, is reflexive, and is contained within every reflexive relation that
contains R, it is called the reflexive closure of R.
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62. CLOSURES OF RELATIONS
• Symmetric Closure
• The symmetric closure of a relation R can be constructed by adding all ordered pairs of the
form (b, a), where (a, b) is in the relation, that are not already present in R.
• Adding these pairs produces a relation that is symmetric, that contains R, and that is
contained in any symmetric relation that contains R.
• The symmetric closure of a relation can be constructed by taking the union of a relation with
its inverse
• The symmetric closure of R is R ⋃ R−1
, where R−1 = {(b, a) | (a, b) ∈ R}.
Tahsin Aziz 62

63. CLOSURES OF RELATIONS
• Example
• The relation {(1, 1), (1, 2), (2, 2), (2, 3), (3, 1), (3, 2)} on {1, 2, 3} is not symmetric.
• To make this a symmetric relation we need to only add (2, 1) and (1, 3), because these are the
only pairs of the form (b, a) with (a, b) ∈ R that are not in R.
• This new relation is symmetric and contains R. Furthermore, any symmetric relation that
contains R must contain this new relation, because a symmetric relation that contains R must
contain (2, 1) and (1, 3). Consequently, this new relation is called the symmetric closure of R.
Tahsin Aziz 63

64. CLOSURES OF RELATIONS
• Transitive Closure
• A transitive relation S containing R such that S is a subset of every transitive relation containing R. Here, S is the
smallest transitive relation that contains R. This relation is called the transitive closure of R.
• A computer network has data centers in boston, chicago, denver, detroit, new york, and san diego.
• There are direct, one-way telephone lines from boston to chicago, from boston to detroit, from chicago to detroit, from
detroit to denver, and from new york to san diego.
• Let R be the relation containing (a, b) if there is a telephone line from the data center in a to that in b. How can we
determine if there is some (possibly indirect) link composed of one or more telephone lines from one center to another?
• Because not all links are direct, such as the link from boston to denver that goes through detroit, R cannot be used
directly to answer this. In the language of relations, R is not transitive, so it does not contain all the pairs that can be
linked.
• We can find all pairs of data centers that have a link by contructing a transitive closure.
Tahsin Aziz 64

65. EQUIVALENCE RELATIONS
• A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.
• Two elements that are related by an equivalence relation are called equivalent.
• The notation a ~ b is often used to denote that a and b are equivalent elements with respect to a
particular equivalence relation.
• Suppose an equivalence relation is transitive, if a and b are equivalent and b and c are equivalent, it
follows that a and c are equivalent.
• Example
• = in general algebra
• ≡ in propositional calculus
Tahsin Aziz 65

66. EQUIVALENCE RELATIONS
• Suppose that R is the relation on the set of strings of English letters such that aRb if and
only if l(a) = l(b), whenever l(x) is the length of the string x. Is R an equivalence relation?
• Solution:
• Let a is a string and aRa which means l(a) = l(a), it follows that R is reflexive.
• Suppose that aRb, then l(a) = l(b). Again let bRa, then l(b) = l(a). Therefore, R is symmetric.
• Let aRb and bRc, then l(a) = l(b) and l(b) = l(c) which imply that l(a) = l(c) that means aRc.
Therefore, R is transitive.
• Because R is reflexive, symmetric, and transitive, it is an equivalence relation.
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67. EQUIVALENCE RELATIONS
• Prove that, “=” is an equivalence relation.
• Solution:
• It is true for a = a. So “=” is reflexive.
• Suppose a = b, then it is also true for b = a. Therefore, “=” is symmetric.
• Suppose a = b and b = c, which show that a = c. So “=” is transitive.
• Because “=” is reflexive, symmetric, and transitive, it is an equivalence relation.
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68. EQUIVALENCE RELATIONS
• Congruence Modulo
• If a and b are two integers and m is an positive integer, then a is congruent to b modulo m if
m divides a − b.
• We use the notation a ≡ b (mod m) to indicate that a is congruent to b modulo m.
• If a and b are not congruent to modulo m, we write a ≢ b (mod m)
• Example: Whether 17 is congruent to 5 modulo 6.
• 17 ≡ 5 (mod 6), 17 − 5 =12 which is divided by 6.
• So we can say 17 is congruent to 5 modulo 6.
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69. EQUIVALENCE RELATIONS
• Congruence modulo m, let m be an integer with m > 1. Show that the relation R = {(a, b) | a ≡ b (mod m)} is
an equivalence relation on the set of integers.
• Solution:
• We know a ≡ b (mod m) if and only if m divides a − b.
• Here a − a = 0 is divisible by m. Hence a ≡ a (mod m), so congruence modulo m is reflexive.
• Next, suppose that a ≡ b (mod m). Then m divides a − b. So a − b = km, where k is an integer. We can write a = b + km. It
follows that b − a = (−k)m means b = a + (-k)m. So b ≡ a (mod m). Hence, congruence modulo m is symmetric.
• Again, suppose that a ≡ b (mod m) and b ≡ c (mod m). Then m divides both a − b and b − c. Therefore, there are integers
k and l with a − b = km and b − c = lm. So we can write a = b + km and c = b - lm. Subtracting c from a we obtain, a - c =
b + km – b + lm. Which equals to km + lm = (k + l)m. Here (k + l) is an integer. Thus, a ≡ c (mod m). Therefore,
congruence modulo m is transitive.
• It follows that congruence modulo m is an equivalence relation.
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70. EQUIVALENCE CLASSES
• Let R be an equivalence relation on a set A. The set of all elements that are related to an
element a of A is called the equivalence class of a.
• The equivalence class of a with respect to R is denoted by [a]R.
• When only one relation is under consideration, the subscript R is often deleted and [a] is
used to denote the equivalence class.
• These classes are disjoint.
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71. EQUIVALENCE CLASSES
• If R is an equivalence relation on a set A, the equivalence class of the element a is
• [a]R = {s | (a, s) ∈ R}.
• If b ∈ [a]R, then b is called a representative of this equivalence class.
• Any element of a class can be used as a representative of this class. That is, there is
nothing special about the particular element chosen as the representative of the class.
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72. EQUIVALENCE CLASSES
• Example
• We can group all the numbers that are equivalent to each other.
• 0 ≡ 2 ≡ 4 ≡ …………(mod 2) and 1 ≡ 3 ≡ 5 ≡ …………(mod 2)
• We can write as
• [0] = {0, 2, 4, ………..} and [1] = {1, 3, 5, ………..}
• [0] and [1] is known as equivalence class.
• 0, 1 are representatives of these equivalence class.
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73. EQUIVALENCE CLASSES
• Consider the relation R = {(a, b) | a ≡ b (mod 2)}. Now find out the equivalence classes from it.
Or,
• What are the equivalence classes of 0 and 1 for congruence modulo 2?
• Solution:
• The equivalence class of 0 contains all integers a such that a ≡ 0 (mod 2). The integers in this class are those
divisible by 2. Hence, the equivalence class of 0 for this relation is [0]={……., −4, −2, 0, 2, 4, …….}.
• The equivalence class of 1 contains all the integers a such that a ≡ 1 (mod 2). The integers in this class are those
that have a remainder of 1 when divided by 2. Hence, the equivalence class of 1 for this relation is [1]={……, -5,
−3, −1, 1, 3, 5, ,…….}.
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74. EQUIVALENCE CLASSES
• If R is an equivalence relation on a set A then there are A1, A2, A3,……. such that
• Ai ⊆ A
• ∀ 𝑎, 𝑏 ∈ 𝐴𝑖
𝑎𝑅𝑏
• Ai ∩ Aj = ∅, when i ≠ j
• A1 ∪ A2 ∪ A3 ∪ ……….. = A
• Here, Ai is an equivalence class. {A1, A2, A3,……. } is a partition of A. e, for e ∈ Ai is a
representative of Ai
• Representative of different classes are not equivalent to each other.
• An equivalence relation on A partitions A into disjoint non- empty sets.
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75. PARTITIONS
• Consider the relation r = {(a, b)| a ≡ b mod 2}.
• This splits the integers into two equivalence classes: even numbers and odd numbers.
• These two sets together form a partition of the integers.
• A partition of a set is a collection of non-empty disjoint subsets of S whose union in S.
• In this example, the partition is {[0], [1]} or {{….., -3, -1, 1, 3,…..}, {….., -4, -2, 0, 2, 4,…..}}
• We have partitioned Z into equivalence classes [0] and [1], under the relation of
congruence modulo 2 = set of integers = {….., -4, -3, -2, -1, 0, 1, 2, 3, 4,…..}
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76. ORDER OF RELATION
• Partial order:
• A binary relation R on a set is a partial order if and only if it is reflexive, antisymmetric and
transitive.
• Total order:
• A binary relation R on a set A is a total order if and only if it is a partial order and for any pair of
elements a and b of A, (a, b) ∈ R or (b, a) ∈ R.
• That is every element is related with every element one way or the other.
• A total order is also called a linear order.
• Quasi order:
• A binary relation R on a set A is a quasi order if and only if it is irreflexive and transitive.
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77. THANK YOU ALL
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