Partially and Totally Ordered Set POSET

In this class, We discuss Partially and Totally Ordered Set POSET.

The reader should have prior knowledge of equivalence relations. Click Here.

Partial Ordered Set or POSET:

A relation is partially ordered if it satisfies 1) Reflexive, 2) Anti-symmetric, and 3) Transitive.

Example:

A = {a, b}

S = ρ(s) = {Φ, {a}, {b}, {a,b}}

Relation R = [S, ⊆]

The relation R is reflexive. because any element is a subset of itself.

The relation R is anti-symmetric. because if x ⊆ y and y ⊆ x then x = y

Anti-symmetric from the properties of the subset.

The relation R is transitive. So, the relation is partially ordered.

Totally ordered set:

POSET is a totally ordered set if every pair of elements has a relation.

First, we check for POSET.

If it is a POSET, then check for the totally ordered set.

The above POSET example {a}, {b} does not form a relation.

In a totally ordered set, every pair should be involved in the relation.

Example:

A = set of real numbers

Relation R = [A, <=]

In the relation R, take any two elements from set A, and we have a relation.

The relation R is satisfying POSET. So, we call relation R a totally ordered set.