Relations / Partial Orders
Least You Need to Know: Partial Orders
A partial order is reflexive, antisymmetric, and transitive. Unlike equivalence relations, not every pair must be comparable.
Least you need to know
- A partial order is reflexive, antisymmetric, and transitive.
- Antisymmetric is different from asymmetric.
- A partial order does not require every pair to be comparable.
- Divisibility and subset are standard examples of partial orders.
- Hasse diagrams omit reflexive and transitive edges.
Key notation
- ≤ — typical partial-order notation
- a | b — a divides b
- A ⊆ B — A is a subset of B
Worked example
- On the power set of
{1,2}, use subset inclusion.\n- Every set is a subset of itself, so the relation is reflexive.\n- IfA⊆BandB⊆A, thenA=B, so it is antisymmetric.\n- Subset inclusion is also transitive, so it is a partial order.
Common mistakes
- Students often mix up antisymmetric and symmetric.
- A missing comparison does not break a partial order.
- Hasse diagrams keep only cover relations, not every implied edge.
How to recognize it
- The prompt asks about comparability, maximal elements, or Hasse diagrams.
- The relation is based on inclusion or divisibility.
- The properties reflexive/antisymmetric/transitive appear together.
Next recommended lesson
Continue through this topic with Least You Need to Know: Minimal, Maximal, Least, and Greatest.
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