Consider the subset relation ⊆ on the power set of {1, 2, 3}. What is the relationship between {1, 2} and {2, 3}?
A{1, 2} ⊆ {2, 3}, because both sets contain 2
B{2, 3} ⊆ {1, 2}, because {1, 2} has smaller-numbered elements
C{1, 2} and {2, 3} are incomparable — neither is a subset of the other
D{1, 2} = {2, 3}, because they have the same cardinality
{1, 2} ⊄ {2, 3} because 1 ∈ {1, 2} but 1 ∉ {2, 3}. Likewise {2, 3} ⊄ {1, 2} because 3 ∈ {2, 3} but 3 ∉ {1, 2}. Neither set is contained in the other, so they are incomparable — neither comes 'before' the other under ⊆. This is precisely what makes subset inclusion a *partial* order rather than a total order.
Question 2 Multiple Choice
Which of the following relations fails to be a partial order, and why?
A≤ on real numbers: for any two reals, one is ≤ the other
B⊆ on sets: subset inclusion is reflexive, antisymmetric, and transitive
C'Has the same number of elements as' on finite sets
DDivisibility on positive integers: a | b means a divides b
'Has the same number of elements as' is an equivalence relation, not a partial order. It violates antisymmetry: two distinct sets can each have the same cardinality as the other (e.g., {1, 2} and {3, 4}) without being equal. Antisymmetry requires that if a ≤ b and b ≤ a, then a = b — but here both sets 'dominate' each other while remaining distinct.
Question 3 True / False
In a partial order, it is possible for two distinct elements a and b to satisfy neither a ≤ b nor b ≤ a.
TTrue
FFalse
Answer: True
This is exactly what 'partial' means. A partial order only requires that the order be consistent when comparisons are made — it does not require every pair to be comparable. Elements that are neither ≤ the other are called incomparable. Total (linear) orders are the special case where every pair is comparable, but most naturally occurring orders (subsets, divisibility, logical implication) have incomparable pairs.
Question 4 True / False
Any relation that is reflexive and transitive is a partial order.
TTrue
FFalse
Answer: False
Antisymmetry is also required. Without it, you have a preorder (or quasi-order), not a partial order. The difference matters: a preorder allows two distinct elements to satisfy a ≤ b and b ≤ a simultaneously, making them 'equivalent' without being identical. Partial orders add antisymmetry to rule this out: if a ≤ b and b ≤ a, then a = b.
Question 5 Short Answer
What does it mean for two elements to be 'incomparable' in a partial order? Give a concrete mathematical example and explain why the existence of incomparable elements is what makes partial orders 'partial.'
Think about your answer, then reveal below.
Model answer: Two elements a and b are incomparable if neither a ≤ b nor b ≤ a holds. For example, under divisibility on positive integers, 3 and 4 are incomparable: 3 does not divide 4, and 4 does not divide 3. The order is 'partial' because it only ranks some pairs of elements — it gives a coherent hierarchy wherever comparisons apply, but leaves other pairs unranked. A total order would require every pair to be comparable.
The existence of incomparable elements captures real hierarchical structure that linear (total) orders cannot express. In mathematics, this appears in subset inclusion, divisibility, logical implication, and the refinement of partitions. Hasse diagrams make incomparability visible: elements on different 'branches' with no connecting path are incomparable.