Day 03Part 13-Example based on lattice in discrete mathematics with all the basic concept.

Day 03Part 13-Example based on lattice in discrete mathematics with all the basic concept.

 Examples Based on Lattice in Discrete Mathematics

 Recap: What is a Lattice?

A lattice is a partially ordered set (poset) (L, ≤) where every pair of elements has:
Least Upper Bound (LUB) → Join ( ∨ )
Greatest Lower Bound (GLB) → Meet ( ∧ )

Graphically, lattices are represented using Hasse diagrams.

 Example 1: Check if a Poset Forms a Lattice

Given Set: $L=\{1,2,3,6\}$ under divisibility relation (|).

 We check LUB (Join) and GLB (Meet) for each pair.

Elements (a, b)	Join (a ∨ b) = LCM	Meet (a ∧ b) = GCD
(1,2)	LCM(1,2) = 2	GCD(1,2) = 1
(2,3)	LCM(2,3) = 6	GCD(2,3) = 1
(2,6)	LCM(2,6) = 6	GCD(2,6) = 2
(3,6)	LCM(3,6) = 6	GCD(3,6) = 3

Since every pair has a Join ( ∨ ) and Meet ( ∧ ), this forms a lattice.

Hasse Diagram Representation:

 Example 2: Complemented Lattice Check

A lattice (L, ∨, ∧) is complemented if:

• It is bounded (Has Top (1) and Bottom (0)).
• Every element a has a complement a’ such that:
  • a ∨ a’ = 1
  • a ∧ a’ = 0

Boolean Algebra Example:
Elements {0,1} form a complemented lattice since:

• 0’ = 1, 1’ = 0
• 0 ∨ 1 = 1, 0 ∧ 1 = 0

Hasse Diagram Representation:

 Example 3: Distributive Lattice Check

A lattice is distributive if it satisfies:

• a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
• a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

Power Set Example: $P(S)=\{\varnothing ,\{a\},\{b\},\{a,b\}\}$

• Operations:
  • Join ( ∨ ) = Union ( ∪ )
  • Meet ( ∧ ) = Intersection ( ∩ )

 Since power sets satisfy distributive laws, they form a distributive lattice.

Hasse Diagram Representation:

 Conclusion

Lattice: Poset where each pair has Join ( ∨ ) & Meet ( ∧ ).
Complemented Lattice: Has Top, Bottom & Complements.
Distributive Lattice: Satisfies Distributive Laws.

Would you like more solved problems or real-life applications of lattices?

Day 03Part 13-Example based on lattice in discrete mathematics with all the basic concept.

Discrete Mathematics for Computer Science

Notes on Lattice Theory

Here’s a detailed explanation for:

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Day 03 – Part 13: Example-Based Learning on Lattices (Discrete Mathematics)

Topic: Example-Based Concepts of Lattices in Discrete Mathematics

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Basic Concept Recap: What is a Lattice?

A lattice is a special kind of partially ordered set (poset) where:

For any two elements a and b, both:

• Join (Least Upper Bound) → a ∨ b

• Meet (Greatest Lower Bound) → a ∧ b

must exist and be unique.

This means: A lattice allows for combining and comparing any two elements using ∨ and ∧ operations.

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Example 1: Set of Divisors of 12

Let S = {1, 2, 3, 4, 6, 12}

Relation: a ≤ b if a divides b
This forms a partial order

Let’s Test Lattice Conditions:

Pick any two elements:

• 2 and 3

  • GCD = 1 → meet (2 ∧ 3) = 1

  • LCM = 6 → join (2 ∨ 3) = 6

• 4 and 6

  • GCD = 2 → meet = 2

  • LCM = 12 → join = 12

All such pairs satisfy meet and join → So S is a lattice.

Meet = GCD
Join = LCM

Hasse Diagram:

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Example 2: Power Set of {a, b}

Let P = P({a, b}) = {∅, {a}, {b}, {a,b}}

Operations:

• Join = Union ( ∪ )

• Meet = Intersection ( ∩ )

Every pair has union and intersection
So this is a lattice

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Distributive and Complemented?

• Distributive: Yes, power sets are always distributive

• Complemented: Yes, each subset has a complement

Hence, this is a Boolean lattice

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Quick Properties Summary

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Practice Questions

1. Is the set {1, 2, 3, 5, 15} under divisibility a lattice?

2. Draw Hasse diagram of power set P({x, y, z}) and show if it’s a Boolean lattice.

3. Show with example whether lattice {0, 1, 2} with min/max forms distributive lattice.

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Need More Help?

I can send:

• PDF Notes of these examples

• Practice MCQs and solved answers

• Visual Hasse Diagrams (PNG/PDF)

Would you like me to prepare a diagram worksheet or animated explanation as well?

Day 03Part 13-Example based on lattice in discrete mathematics with all the basic concept.