🔢 Boolean Algebra Explained from Scratch

Welcome to the fascinating world of Boolean Algebra! 👋

In this guide, we'll explore:

What is Boolean Algebra?
Boolean Operators
Boolean Laws and Theorems

Did you know?

Boolean algebra is the foundation of all modern digital computers! 💻

What is Boolean Algebra?

Boolean algebra is a branch of mathematics that works with only two values:

0 → False (OFF, Low voltage) 🔌
1 → True (ON, High voltage) 💡

It was introduced by George Boole in the 19th century.

In computers, Boolean algebra is the foundation of digital logic (how circuits and processors work). 🔧

Example:

A simple light switch:

Switch OFF = 0
Switch ON = 1

If we say Light = Switch1 AND Switch2, the light will only turn ON if both switches are ON. 💡

Boolean Operators

Boolean algebra uses operators just like normal math does (+, -, ×), but here they mean something different:

🔗

AND (•)

Output is 1 only if both inputs are 1.

🔀

OR (+)

Output is 1 if at least one input is 1.

🔄

NOT (¬ or ')

Flips the input (0 → 1, 1 → 0).

Visual Example:

AND: Like two switches in series - both must be ON for the light to work. 💡

OR: Like two switches in parallel - either one can turn the light ON. 💡

NOT: Like an inverter - it flips the state. 🔄

Boolean Laws and Theorems - Part 1

These laws are like the rules of arithmetic, but for logic. They make it possible to simplify complex logical circuits. ⚙️

Commutative Laws

Order doesn't matter.

OR: A + B = B + A

AND: AB = BA

Example:

Pressing Switch A OR B will turn ON the fan, no matter the order. 🌀

Associative Laws

Grouping doesn't matter.

OR: (A + B) + C = A + (B + C)

AND: (AB)C = A(BC)

Example:

If 3 people vote YES, the result doesn't depend on the grouping order. 👥

Distributive Laws

Just like in normal algebra, but with AND/OR.

OR distributes over AND: A(B + C) = AB + AC

AND distributes over OR: A + BC = (A + B)(A + C)

Example:

A(B + C) → If A must be ON and either B or C is ON → it's same as (A and B) OR (A and C). 💡

Boolean Laws and Theorems - Part 2

Identity Laws

Adding or multiplying by constants 0 or 1.

OR: A + 0 = A

AND: A • 1 = A

OR: A + 1 = 1

AND: A • 0 = 0

Example:

Anything OR 1 = always 1 (a lamp connected to a permanent battery is always ON). 🔋

Complement Laws

A variable plus its opposite covers all cases.

A + A′ = 1

A • A′ = 0

Example:

A switch and its opposite cannot be ON together. 🔌

A	A′	A + A′ (OR)	A • A′ (AND)
0	1	0 + 1 = 1	0 × 1 = 0
1	0	1 + 0 = 1	1 × 0 = 0

Boolean Laws and Theorems - Part 3

Involution Law

Double negation brings you back to the original value:

(A′)′ = A

This means: If you take the complement (NOT) of a variable twice, you get the original variable back.

What does this mean with numbers?

Recall:

If A = 0, then A′ = 1
If A = 1, then A′ = 0

Now apply double negation:

A	A′	(A′)′	Result compared to A
0	1	0	Same as A = 0
1	0	1	Same as A = 1

Real-world example:

Imagine a switch A:

A′ means NOT A (switch OFF if A was ON, ON if A was OFF).

Taking NOT again, (A′)′, returns it back to the original state of A.

So:

NOT(NOT(ON)) = ON
NOT(NOT(OFF)) = OFF

Summary: The double NOT cancels out and leaves the original value unchanged. This is similar to saying: "The opposite of the opposite is the original." 🔄