Relations: Difference between revisions
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'''Theorem: An equivalence relation is the same as a partition'''. The equivalence classes of any equivalence relation R on a set A for a partition of A, and any partition of A determines an equivalence relation on A for which the sets in the partition are the equivalence classes. | '''Theorem: An equivalence relation is the same as a partition'''. The equivalence classes of any equivalence relation R on a set A for a partition of A, and any partition of A determines an equivalence relation on A for which the sets in the partition are the equivalence classes. | ||
=Partial Order= | |||
A relation that is [[#Reflexive_Relation|reflexive]], [[#Antisymmetric_Relation|antisymmetric]] and [[#Transitive_Relation|transitive]] is a '''partial order'''. We call a set on which a partial order is defined a '''partially ordered set'''. | |||
For example, the relation "is a descendant of" is a partial order on the set of all people, if we allow that individuals are being their own descendants. | |||
In a partially ordered set, there may be no single "maximum" element a such that ''b R a'' for all b ∈ A. Instead, the set may contain several '''maximal''' elements a such that for no b ∈ A, where b ≠ a, is it the case that ''a R b''. | |||
=TODO= | =TODO= | ||
<font color=darkgray>[[CLRS]] page 1163</font> | <font color=darkgray>[[CLRS]] page 1163</font> | ||
Revision as of 22:06, 27 August 2018
Internal
Overview
A binary relation R on two sets A and B is a subset of the Cartesian product A x B. If (a, b) belongs to the subset of the Cartesian product that defines the relation, we write a R b.
A binary relation R on a set A is a subset of the Cartesian product A x A.
A n-ary relation on sets A1, A2, .... An is a subset of the Cartesian product A1 x A2 x ... x An.
An example of a binary relation on a finite set is the edge set of a graph.
Binary Relation Properties
A binary relation R ⊆ A x A is reflexive if a R a for all a ∈ A.
A binary relation R ⊆ A x A is symmetric if a R b implies b R a for all a, b ∈ A.
A binary relation R ⊆ A x A is transitive if a R b and b R c implies a R c for all a, b, c ∈ A.
A binary relation R ⊆ A x A is antisymmetric if a R b and b R a imply a = b. For example, the "≤" relation on natural numbers is antisymmetric, since a ≤ b and b ≤ a imply a = b.
Equivalence
A relation that is reflexive, symmetric and transitive is an equivalence relation. For example, "=" is an equivalence relation on the natural numbers.
Equivalence class. If R is an equivalence relation on the set A, then for a ∈ A, the equivalence class of a is the set [a] = {b ∈ A, where a R b}. In other words, the equivalence set of a is the set of all elements equivalent to a, relative to relation R.
Theorem: An equivalence relation is the same as a partition. The equivalence classes of any equivalence relation R on a set A for a partition of A, and any partition of A determines an equivalence relation on A for which the sets in the partition are the equivalence classes.
Partial Order
A relation that is reflexive, antisymmetric and transitive is a partial order. We call a set on which a partial order is defined a partially ordered set.
For example, the relation "is a descendant of" is a partial order on the set of all people, if we allow that individuals are being their own descendants.
In a partially ordered set, there may be no single "maximum" element a such that b R a for all b ∈ A. Instead, the set may contain several maximal elements a such that for no b ∈ A, where b ≠ a, is it the case that a R b.
TODO
CLRS page 1163