Partial Orderings A partial ordering is a binary relation on a set that is not reflexive. In simpler words, it is a relation where you can go from one elemen...
A partial ordering is a binary relation on a set that is not reflexive. In simpler words, it is a relation where you can go from one element to another, but you cannot go from a specific element back to itself.
Examples:
The relation "is taller than" is a partial ordering on the set of people.
The relation "is a student in this class" is a partial ordering on the set of students.
The relation "is a parent of" is a partial ordering on the set of families.
Properties of Partial Orderings:
A partial ordering is reflexive if and only if the relation is symmetric.
A partial ordering is transitive if and only if the relation is compatible with the composition of sets.
A partial ordering is a partial ordering on a subset of the original set.
Partial Ordering Example:
Consider the set of all people who are residents of New York City. We can define a partial ordering on this set by saying that a person is more senior than another if they were born before them. This relation is not reflexive because it does not satisfy the property that a person can be their own parent.
Key Points:
Partial orderings are a broader class of relations than reflexive orderings.
They are used to model situations where it is not possible to go from one element to another directly.
Partial orderings can be used to define many different structures in discrete mathematics