Day 03Part 10- Discrete mathematics – Introduction of Lattices and and it’s type in Very easy ways.

Day 03Part 10- Discrete mathematics – Introduction of Lattices and and it’s type in Very easy ways.

 Day 03 | Part 10 – Discrete Mathematics

 Introduction to Lattices & Their Types (Easy Explanation)

 What is a Lattice?

A Lattice is a type of algebraic structure in Discrete Mathematics that helps in organizing elements in a structured way. It is a partially ordered set (poset) where every two elements have:

• A Least Upper Bound (LUB) called Supremum or Join (∨)
• A Greatest Lower Bound (GLB) called Infimum or Meet (∧)

Example of a Lattice:
Consider the set {1, 2, 4, 8} with the divisibility relation (|)

• LUB (Join ∨): The smallest number that is divisible by both.
• GLB (Meet ∧): The largest number that divides both.

Join (∨): 2 ∨ 4 = 4 (smallest multiple of both)
Meet (∧): 2 ∧ 4 = 2 (largest common divisor)

 If a poset satisfies both Join & Meet properties, it is a Lattice!

 Types of Lattices

 1. Bounded Lattice

A Lattice with the greatest (1) & least (0) elements.
 Example: {0, 1, 2, 3, 4, 6, 12} under divisibility.

• 0 is the least element (divides all).
• 12 is the greatest element (divisible by all).

 2. Distributive Lattice

A Lattice where Join (∨) and Meet (∧) distribute over each other:
A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)

Example: The set {1, 2, 3, 6} under divisibility is distributive.

 3. Complemented Lattice

A Bounded Lattice with a Complement (¬A) such that:

• A ∨ (¬A) = 1
• A ∧ (¬A) = 0

 Example: The Boolean algebra {0,1} is a Complemented Lattice.

 4. Modular Lattice

A Lattice that follows a weaker distributive law:
If A ≤ C, then A ∨ (B ∧ C) = (A ∨ B) ∧ C

 Example: The subgroup lattice of a group forms a modular lattice.

 Key Differences Between Lattice Types

 Real-Life Applications of Lattices

Boolean Algebra – Used in computer logic circuits & programming
Database Design – Organizing hierarchical data
Cryptography – Lattice-based encryption methods
Artificial Intelligence – Decision-making systems

 Summary

• Lattice = A poset where every pair has Join (∨) & Meet (∧).
• Types: Bounded, Distributive, Complemented, Modular.
• Used in: Logic, AI, Cryptography, and Databases.

Would you like examples or solved problems on lattices?

Day 03Part 10- Discrete mathematics – Introduction of Lattices and and it’s type in Very easy ways.

dmoi-tablet.pdf – Discrete Mathematics

Types of Lattice