1. Introduction to Relations
CSC 333 Discrete Mathematics

2. Terminology Terms binary relation reflexive symmetric transitive
antisymmetric
partial ordering
poset
equivalence relation
closure
total ordering
partition
topological sort
function
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3. Binary Relations
A binary relation is a relationship between elements of the same set.
More formally, see defn, p. 287.
Given the set S = {1,2,4,8,16,32}, ρ is a binary relation, i.e., a subset of S X S such that x ρ y  (x,y) Є ρ.
Thus, {(1,2), (4,8), (16,32)} is a binary relation on set S|x ρ y  x = y/2.
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4. Binary Relations
Note that n-ary relations on multiple sets are possible (p. 287).
Binary relations may be
Reflexive
Symmetric
Transitive
Antisymmetric
(Pages )
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5. Closure
Informally, a binary relation ρ* is the closure of a relation ρ with respect to property P if it contains the minimal set of ordered pairs needed for the property to hold. In other words, suppose the relation ρ on set S does not have the property of reflexivity; if the necessary ordered pairs are added to S in order for reflexivity to hold, then ρ* is closed with respect to reflexivity.
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6. Closure For a more formal definition, see page 291. Note Example 9.
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7. Partial Ordering Requires
Reflexivity
Antisymmetry
Transivity
If ρ is a partial ordering on S, then the ordered pair (S, ρ) is called a poset.
(Page 293)
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8. Partial Ordering
Related concepts: predecessor, successor, immediate predecessor.
A partial ordering in which every element of a set is related to every other element of the set is a total ordering.
(Pages )
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9. Using Hasse Diagrams to Represent Posets
Named after Helmut Hasse ( )
Hasse Diagram – A directed graph .
Can be used to represent a finite poset.
Nodes (vertices) represent elements of the set.
Edges show relationships between nodes.
Edges that must be present by definition are not shown.
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10. Using Hasse Diagrams to Represent Posets
Unrelated elements of the set are shown in the H.D. as unconnected nodes.
Definition: Total Ordering – a partial ordering in which every element of the set is related to every other element. (Fig. 4.4)
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11. Equivalence Relation Requires
Reflexivity
Symmetry
Transivity
Equivalence Class – For the equivalence relation ρ on S and x Є S, the set of all members of S to which x is related, denoted as [x]. (Pages )
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12. An Equivalence Relation
Partitions the set on which it is defined.
Partition: a collection of disjoint subsets of S whose union is S.
See Example 12.
See related theorem.
(Page 296)
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13. Equivalence Classes Congruence Modulo n Definition, page 299.
See Example 15.
See Practice 15.
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