Day 03Part 07-Discretse mathematics for cse-Finding of maxiamal, minimal, maximum, minimum points.

Discrete Mathematics for CSE

Day 03 – Part 07: Finding Maximal, Minimal, Maximum, and Minimum Points

In Discrete Mathematics, finding maximal, minimal, maximum, and minimum points is essential in set theory, lattice theory, and optimization. Let’s break down these concepts:

1. Maximal and Minimal Elements

In a partially ordered set (poset) $(S,\le )$, elements are compared based on their ordering relation.

Maximal Element:

An element $a\in S$ is maximal if there is no element $b\in S$ such that $a<b$.
Example: In the set $S=\{1,3,5,7\}$ ordered by divisibility, 7 is maximal (since no element is greater in divisibility order).

Minimal Element:

An element $a\in S$ is minimal if there is no element $b\in S$ such that $b<a$.
Example: In the same set $S=\{1,3,5,7\}$, 1 is minimal (since no element divides it except itself).

2. Maximum and Minimum Elements

If a maximal (or minimal) element is the greatest (or least) element of the set, it is called a maximum (or minimum).

Maximum Element:

An element $m$ is maximum if $m\ge x$ for all $x\in S$.
Example: In $S=\{1,2,3,4,5\}$, the maximum is 5.

Minimum Element:

An element $m$ is minimum if $m\le x$ for all $x\in S$.
Example: In $S=\{1,2,3,4,5\}$, the minimum is 1.

Key Differences:

Concept	Definition	Uniqueness
Maximal	No element is strictly greater	May not be unique
Minimal	No element is strictly smaller	May not be unique
Maximum	The greatest element in the set	Unique if exists
Minimum	The smallest element in the set	Unique if exists

3. Example in a Partially Ordered Set

Consider the set $S=\{2,4,6,8,10\}$ with divisibility as the order relation:

• Minimal element: $2$ (as nothing divides it except itself).
• Maximal element: $10$ (as no number is divisible by it except itself).
• Maximum: $10$ (since it is the greatest).
• Minimum: $2$ (since it is the smallest).

Would you like me to explain lattice theory applications or provide an algorithm to find these elements?

Day 03Part 07-Discretse mathematics for cse-Finding of maxiamal, minimal, maximum, minimum points.

TNS Theoretical and Natural Science – Advances in Engineering …

Mathematics for Computer Science – courses

Notes on Discrete Mathematics

Maxima and minima

Certainly! Here’s a well-structured explanation for:

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Day 03 Part 07 – Discrete Mathematics for CSE

Finding Maximal, Minimal, Maximum, and Minimum Elements in a Partially Ordered Set

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Key Concepts – Poset (Partially Ordered Set)

A partially ordered set (poset) is a set P with a binary relation ≤ that is:

• Reflexive: a ≤ a
• Antisymmetric: If a ≤ b and b ≤ a, then a = b
• Transitive: If a ≤ b and b ≤ c, then a ≤ c

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Important Definitions

1. Maximal Element:

An element m ∈ P is called maximal if no other element is greater than m.

• There can be multiple maximal elements.
• Not necessarily the greatest in the set.

Formally:
There is no element x ∈ P such that m < x

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2. Minimal Element:

An element m ∈ P is called minimal if no other element is less than m.

• There can be multiple minimal elements.
• Not necessarily the least in the set.

Formally:
There is no element x ∈ P such that x < m

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3. Maximum Element:

An element M ∈ P is called the maximum if it is greater than or equal to all other elements in the set.

Formally:
For all x ∈ P, x ≤ M

There can be only one maximum.

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4. Minimum Element:

An element m ∈ P is the minimum if it is less than or equal to all other elements in the set.

Formally:
For all x ∈ P, m ≤ x

There can be only one minimum.

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Visual Example: Hasse Diagram

Consider a poset P = {a, b, c, d} with the relation:

    a
   / \
  b   c
   \ /
    d

From this:

• Maximal Elements: a (no one above it)
• Minimal Elements: d (no one below it)
• Maximum Element: Not unique (a is not greater than both b and c)
• Minimum Element: Not unique

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Tip for GATE/University Exams:

• Use Hasse Diagrams to easily visualize maximal/minimal points.
• A maximal element is topmost in its branch.
• A minimal element is bottommost in its branch.

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Quick Practice:

Given: P = {1, 2, 3, 4}, with relation R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3)}

Find:

• Maximal elements = ?
• Minimal elements = ?
• Maximum = ?
• Minimum = ?

Answer:

• Maximal = {4, 3}
• Minimal = {1, 4}
• Maximum =
• Minimum =

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Would you like:

• Hasse diagram visuals?
• PDF notes or MCQs?
• Hindi/English video explanation?

Let me know and I’ll send them your way!