DISCO - CS F222
Tutorial 5
September 27, 2025
Q1. Determine whether the following are partial orders or not. If they are partially ordered are
they strict?:
(a) S = Z+ , xRy ⇐⇒ x | y
(b) S = R, xRy ⇐⇒ |x| ≤ |y|
(c) S = R2 , (x1 , y1 )R(x2 , y2 ) ⇐⇒ x21 + y12 < x22 + y22
(d) Dual poset of (S, ⪯) where S is any finite set
Q2. Prove that in a poset (S, ⪯), both the maximum and minimum element are unique (Prove
it considering the case that they do exist).
Q3. Let S = {a,b,c,d} and define a partial order ⪯ on S on the basis that a ≤ x ∀ x ∈ S and
no other elements are comparable.
(a) Prove that (S, ⪯) is a poset.
(b) Identify all maximal and minimal elements
(c) For each pair {x, y} ⊂ S, determine if a supremum and infimum exist.
Q4. Prove that a poset is a total order iff every 2-element subset has both an infimum and
supremum which are both contained within the subset itself. Provide an example of a situation
where the infimum and supremum exist but need not be contained within the subset.
Q5. Show that every total ordered poset is a lattice poset.
Q6. Let ⪯=⪯1 ∩ ⪯2 , where ⪯1 and ⪯2 are partial orderings on a set X.
Suppose an element y covers x in the poset (X, ⪯); that is, x ≺ y and there is no z ∈ X such
that x ≺ z ≺ y in ⪯.
Must y cover x in at least one of the original partial orderings ⪯1 or ⪯2 ? Prove or provide a
counterexample.
Q7. Show that any finite poset has atleast one linear extension. Subsequently, derive a relation
between the number of possible topological sorts and the number of linear extensions.
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