Day 03Part 07-Discretse mathematics for cse-Finding of maxiamal, minimal, maximum, minimum points.
Day 03Part 07-Discretse mathematics for cse-Finding of maxiamal, minimal, maximum, minimum points.
Contents [hide]
- 1 Discrete Mathematics for CSE
- 2 Day 03 – Part 07: Finding Maximal, Minimal, Maximum, and Minimum Points
- 3 1. Maximal and Minimal Elements
- 4 Maximal Element:
- 5 Minimal Element:
- 6 2. Maximum and Minimum Elements
- 7 Maximum Element:
- 8 Minimum Element:
- 9 Key Differences:
- 10 3. Example in a Partially Ordered Set
- 11 Day 03Part 07-Discretse mathematics for cse-Finding of maxiamal, minimal, maximum, minimum points.
- 12 TNS Theoretical and Natural Science – Advances in Engineering …
- 13 Mathematics for Computer Science – courses
- 14 Notes on Discrete Mathematics
- 15 Maxima and minima
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,≤)(S, \leq), elements are compared based on their ordering relation.
Maximal Element:
An element a∈Sa \in S is maximal if there is no element b∈Sb \in S such that a<ba < b.
Example: In the set S={1,3,5,7}S = \{1, 3, 5, 7\} ordered by divisibility, 7 is maximal (since no element is greater in divisibility order).
Minimal Element:
An element a∈Sa \in S is minimal if there is no element b∈Sb \in S such that b<ab < a.
Example: In the same set S={1,3,5,7}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 mm is maximum if m≥xm \geq x for all x∈Sx \in S.
Example: In S={1,2,3,4,5}S = \{1, 2, 3, 4, 5\}, the maximum is 5.
Minimum Element:
An element mm is minimum if m≤xm \leq x for all x∈Sx \in S.
Example: In S={1,2,3,4,5}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}S = \{2, 4, 6, 8, 10\} with divisibility as the order relation:
- Minimal element: 22 (as nothing divides it except itself).
- Maximal element: 1010 (as no number is divisible by it except itself).
- Maximum: 1010 (since it is the greatest).
- Minimum: 22 (since it is the smallest).
Would you like me to explain lattice theory applications or provide an algorithm to find these elements?