Here is a clear, complete, and exam-oriented explanation of Partial Order Relations in Discrete Structures. Useful for BCA/MCA/University exams.

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⭐ Partial Order Relations

A relation ( R ) on a set ( A ) is called a partial order relation if it satisfies three properties:

1. Reflexive
2. Antisymmetric
3. Transitive

Such a relation is also called a partial ordering, and the pair ((A, R)) is called a partially ordered set or poset.

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⭐ Definition

A relation ( R \subseteq A \times A ) is a partial order if:

1. Reflexive:
   [
   (a,a) \in R \quad \forall a \in A
   ]
2. Antisymmetric:
   [
   (a,b) \in R \text{ and } (b,a) \in R \Rightarrow a = b
   ]
3. Transitive:
   [
   (a,b) \in R \text{ and } (b,c) \in R \Rightarrow (a,c) \in R
   ]

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⭐ Examples of Partial Order Relations

✔ 1. “Less than or equal to” (≤) on real numbers

• Reflexive: a ≤ a
• Antisymmetric: if a ≤ b and b ≤ a → a = b
• Transitive: if a ≤ b and b ≤ c → a ≤ c

Thus, ≤ is a partial order.

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✔ 2. Subset relation (⊆) on set of all subsets

Let A = {1,2,3}.
Power set P(A) = {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

Relation:
[
B \subseteq C
]

• Reflexive: every set is a subset of itself
• Antisymmetric: if B ⊆ C & C ⊆ B → B = C
• Transitive: B ⊆ C & C ⊆ D → B ⊆ D

Thus, ⊆ is a partial order.

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✔ 3. Divisibility relation (|) on positive integers

a | b means “a divides b”.

Example on A = {1,2,4,8}.

• Reflexive: a divides itself
• Antisymmetric: if a | b and b | a → a = b
• Transitive: if a | b and b | c → a | c

So “divides” is a partial order.

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⭐ Non-Examples (NOT Partial Orders)

✘ 1. < (strict less than)

Not reflexive.

✘ 2. Symmetric relations

Antisymmetry fails.

✘ 3. Congruence modulo n

Not antisymmetric.

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⭐ Poset (Partially Ordered Set)

A set A together with a partial order R is a poset, written as:

[
(A, R)
]

Examples:

• ((\mathbb{Z}, \le))
• ((\mathcal{P}(A), \subseteq))
• (({1,2,4,8}, \mid))

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⭐ Hasse Diagram

A Hasse diagram is a graphical representation of a partial order.

Rules:

• Draw nodes for elements
• Draw an upward edge from a to b if a < b and no element c satisfies a < c < b
• Remove reflexive/loop edges
• Transitive edges are omitted

Example: Poset ({1,2,4,8}) under divisibility

    8
    |
    4
    |
    2
    |
    1

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⭐ Partial Order vs Total Order

A partial order becomes a total order if:

[
\forall a,b \in A, \quad aRb \text{ or } bRa
]

That means every pair is comparable.

Example of Total Order:

(ℝ, ≤)

Example of Partial But NOT Total:

Power set under ⊆
{1} and {2} are not comparable.

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⭐ Quick Exam Answer Summary

Partial Order Relation:
A relation that is:

Property	Meaning
Reflexive	(a,a) ∈ R
Antisymmetric	(a,b),(b,a) ∈ R → a=b
Transitive	(a,b),(b,c) ∈ R → (a,c)

Examples:

• ≤ on numbers
• ⊆ on sets
• Divides (|)
• Alphabet ordering (a ≤ b ≤ c…)

Poset: (A, R)

Hasse Diagram used to visualize partial orders.

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