We will explore abstract algebra concepts step by step using Boolean Algebra as our case study. 🎲✨

What is Boolean Algebra?

🧩 The Light Switch Game

🎯 Understand how Boolean Algebra models binary operations.

💡 You enter a mysterious room with two light bulbs 💡💡 and two switches 🔳🔳.

• Each switch has two states: ON (1)[🔲] or OFF (0)[🔳].

• The room follows a strange logic:

  1. If either switch is ON, the red light 🔴 turns ON.
  2. If both switches are ON, the green light 🟢 turns ON.
  3. The lights turn off if these conditions don't apply.

🤔 Question: What kind of operation describes this behavior?

📌 Defining the Boolean Set

The light switch system follows a binary structure:

\color{gold}
B=｛0,1｝

where:

• 🔳 0 means OFF,
• 🔲 1 means ON.

✅ Boolean algebra operates on just two values: $｛0,1｝$ .

📌 Boolean Operations

The room follows two main rules:

• 1️⃣ 🔴 The "OR" Switch (Logical OR $∨$ )
  • The light turns on if at least one switch is ON🔲.
• 2️⃣ 🟢 The "AND" Switch (Logical AND $∧$ )
  • The light turns on only if both switches are ON🔲.

🔳 Switch 1	🔳 Switch 2	🔴 Light (A OR B)	🟢 Light (A AND B)
🔳 0	🔳 0	⚫ 0	⚫ 0
🔳 0	🔲 1	🔴 1	⚫ 0
🔲 1	🔳 0	🔴 1	⚫ 0
🔲 1	🔲 1	🔴 1	🟢 1

📌 Boolean Algebra as a Lattice

🧩 The Hierarchy of Truth

🎯 Understand how Boolean Algebra forms a lattice, a fundamental structure in algebra.

📖 The Hierarchy of Truth 📜

📌 What is a Lattice?

A lattice is a set where every pair of elements has:

• 🔺 A Greatest Lower Bound (GLB, also called meet, denoted $⋀$)
• 🔻 A Least Upper Bound (LUB, also called join, denoted $⋁$ )

In Boolean Algebra:

• $A∧B$ gives the "greatest" truth that is common to both statements.
• $A∨B$ gives the "least" truth that includes both statements.

📒 Question:

• What happens when you combine the statements "It is raining" and "It is cloudy" using AND $(∧)$? What happens when you combine them using OR $(∨)$?

Each statement (A, B, etc.) forms part of a hierarchical structure, where:

• $0$ (false) is the lowest element.

• $1$ (true) is the highest element.

• $A∨B$ is the lowest truth that contains A and B.

• $A∧B$ is the highest truth that is contained in both A and B.

• 📒 Check if $A∧B=B∧A$ (Commutativity).

• 📒 Check if $A∧(B∧C)=(A∧B)∧C$ (Associativity).

Why Boolean Algebra Forms a Lattice

Boolean Algebra satisfies lattice properties because:

• 1️⃣ It is a set B={0,1}B={0,1} with two operations $∧$ and $∨$.
• 2️⃣ Every pair of elements has a GLB (meet) and LUB (join).
• 3️⃣ It satisfies
  • Associativity: $\frac{A∧(B∧C)=(A∧B)∧C}{A∨(B∨C)=(A∨B)∨C}$
  • Commutativity: $\frac{A∧B=B∧A }{A∨B=B∨A }$
  • Idempotency: $\frac{A∧A=A }{A∨A=A }$

This is a fundamental structure in logic, computing, and abstract algebra!

📌 Boolean Algebra as a Group?

A group is a set with a single binary operation that satisfies:

• Closure: $a∗b$ is still in the set.
• Associativity: $(a∗b)∗c=a∗(b∗c)$.
• Identity Element: There exists an element $e$ such that $a∗e=a$.
• Inverses: Every element has an inverse $a^{−1}$ such that $a∗a^{−1}=e$.

🔳 A	🔳 B	🔴 A OR B	🟢 A AND B	A XOR B
🔳 0	🔳 0	⚫ 0	⚫ 0	⚫ 0
🔳 0	🔲 1	🔴 1	⚫ 0	🟡 1
🔲 1	🔳 0	🔴 1	⚫ 0	🟡 1
🔲 1	🔲 1	🔴 1	🟢 1	⚫ 0

• AND:

  • Closure ✅ (only 0 and 1 appear)
  • Associativity ✅ $(A∧B)∧C=A∧(B∧C)$
  • Identity? ❌ There is no element $e$ such that $A∧e=A$ for all $A$.
  • Inverses? ❌ There is no inverse for $1$ (since $1∧x=1$ has no solution for $x$).
  • 🚨 Boolean Algebra under AND is NOT a group.

• OR:

  • Closure ✅ (only $0$ and $1$ appear)
  • Associativity ✅ $(A∨B)∨C=A∨(B∨C)$
  • Identity? ❌ There is no element $e$ such that $A∨e=A$ for all $A$.
  • Inverses? ❌ No inverse for $0$ (since $0∨x=0$ has no solution for $x$).

• XOR

  • ✅ Closure

  • ✅ Associativity

  • ✅ Identity: $0$ since $A⊕0$ = A$

  • ✅ Inverses: $A⊕A=0$ (so every element is its own inverse!)

  • 🔥 Boolean Algebra under XOR is a Group! In fact, it forms an Abelian group of order 2 (isomorphic to $ℤ₂$​).