Elements of POSET Practice Questions

The elements of a POSET are the individual objects or members of a set that are arranged according to a partial order relation. This relation compares the elements and must satisfy three properties: reflexivity, anti-symmetry, and transitivity.

Question 1: For a set A = {1, 2, 3}, check if the following relations are POSET?

• R₁ = {(1, 1), (2, 2), (3, 3)}
• R₂ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}
• R₃ = { }

To prove a Partial Order Relation, check Reflexivity, Anti-Symmetry and Transitivity.

(1, 1)	(1, 2)	(1, 3)
(2, 1)	(2, 2)	(2, 3)
(3, 1)	(3, 2)	(3, 3)

For Relation R₁

• Reflexive: Since (1, 1) (2, 2) (3, 3) are present so it is Reflexive.
• Anti-symmetry: It allows reflexive pairs, so it is Anti-symmetric.
• Transitive: Reflexive pairs are always Transitive.

For Relation R₂

• Reflexive: Since (1, 1) (2, 2) (3, 3) are present so it is Reflexive.
• Anti-symmetry: For (1, 2) there is (2, 1) so not Anti-symmetric.
• Transitive: There are no such pairs (a, b) (b, c) such that (a, c) is not present.

For Relation R₃

• Reflexive: NULL sets doesn't contain either of (1, 1) (2, 2) (3, 3).

Therefore, R₁ is a POSET, but R₂ and R₃ are not.

Question 2: In the POSET (P(A), ⊆) where A = {a, b, c} and P(A) is the power set of A, identify comparable and incomparable pairs.

Solution:

Comparable pairs: (∅, {a}), (∅, {b}), (∅, {c}), ({a}, {a,b}), ({b}, {a,b}), etc.

Incomparable pairs: ({a}, {b}), ({a}, {c}), ({b}, {c}), ({a,b}, {a,c}), etc.

Question 3: In the POSET (P({a, b, c}), ⊆), identify a maximal chain and a maximal antichain.

Solution:

A maximal chain: ∅ ⊂ {a} ⊂ {a,b} ⊂ {a,b,c}

A maximal antichain: {{a}, {b}, {c}}

Question 4: Find a linear extension of the POSET ({1, 2, 3, 4}, R) where R = {(1, 2), (1, 3), (2, 4), (3, 4)}.

Solution:

A linear extension is a total order that respects the partial order.

One possible linear extension is: 1 < 2 < 3 < 4

Another valid linear extension is: 1 < 3 < 2 < 4

Question 5: In the Boolean algebra of subsets of {1, 2, 3}, find the complement of {1, 3}.

Solution:

The universe U = {1, 2, 3}

The complement of {1, 3} is U \ {1, 3} = {2}

Question 6: In the lattice of divisors of 30, find:

• 6 ∨ 10 (join)
• 6 ∧ 10 (meet)

Solution:

Divisors of 30: {1, 2, 3, 5, 6, 10, 15, 30}

• 6 ∨ 10 = LUB(6, 10) = 30
• 6 ∧ 10 = GLB(6, 10) = 2

Practice Problems

Example 1: Determine if the POSET ({1, 2, 3, 6, 12, 18}, |), where | denotes "divides," is a lattice.

Example 2: Given the POSET P = ({a, b, c, d}, R) where R = {(a, b), (a, c), (b, d), (c, d)}, find its dual POSET.

Example 3: In the POSET ({1, 2, 3, 4, 5, 6}, ≤), find the upper and lower bounds of the subset {2, 3, 5}.

Example 4: Consider the POSET (P, ≤) where P = {1, 2, 3, 4} and ≤ is the "divides" relation.

Example 5: Draw the Hasse diagram for the POSET ({1, 2, 3, 4, 6, 12}, |) where | denotes "divides".