A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on.
Table of Contents
What is a Partial Order Relation?
A relation R on a set A is called a partial order relation if it is
- Reflexive Relation: (a, a) ∈ R ∀ a ∈ A, i.e. aRa for all a ∈ A.
- Anti-Symmetric Relation: ∀ a, b ∈ A, (a, b) ∈ R then (b, a) ∉ R OR a = b.
- Transitive Relation: ∀ a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R.
where R is a subset of (A x A), i.e. the cartesian product of set A with itself.
Example:
Consider set A = {a, b}
R = {(a, a), (b, b), (a, b), (b, a)} is not partial order as for tuple (a, b), (b, a) tuple is present but
R = {(a, a), (a, b), (b, b)} is a partial order relation.
Properties of Partial Order Relation:
The few properties of a partial order relation are:
- Empty relation on a non-empty set is never partial order.
- Universal relation over a non-empty set is never a partial order.
- The smallest partial-order relation will only contain aRa tuples.
How to verify a Partial Order Relation?
The process of identifying/verifying if any given relation is a partial order relation is:
- Check if the relation is Reflexive.
- Check if the relation is Anti-Symmetric.
- Check if the relation is Transitive.
Follow the below illustration for a better understanding
Illustration:
Consider set R = {(1, 1), (1, 3), (1, 4), (2, 2), (2, 1), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4), (4, 3)}
Pairs (1, 1), (2, 2), (3, 3), (4, 4) exist:
⇒ This satisfies the reflexive condition.The transitive condition is also satisfied.
For the pairs (4, 3):
⇒ The relation (3, 4) exists
⇒ This does not satisfy the anti-symmetric condition.
So the relation is not anti-symmetric.Hence it is not a partial order relation.
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