🔢 Boolean Algebra Simplification

Learn how to simplify Boolean expressions with examples

Example 1: Simplify (A+B)(A+C)

We are going to simplify this Boolean expression to make it shorter and more efficient.

Step 1: Apply Distributive Law

Just like in algebra:

(X+Y)(Z+W) = XZ + XW + YZ + YW

So here:

(A+B)(A+C) = A(A+C) + B(A+C)

This is called distribution over OR:

(X+Y)Z = XZ + YZ
Step 2: Expand both terms

Now expand both parts:

= A·A + A·C + B·A + B·C

Which gives:

= A² + AC + AB + BC

(Note: A·A = A², and in Boolean algebra, we use • for AND.)

Step 3: Apply the Idempotent Law

In Boolean algebra:

A·A = A

So:

A² = A

Now the expression becomes:

A + AC + AB + BC
Step 4: Apply the Absorption Law

One of the key rules:

A + AB = A and A + AC = A

So:

A + AC = A
A + AB = A

Apply both:

A + AC + AB + BC ⇒ A + BC
✅ Final Simplified Expression: A + BC

💡 Why simplify?

The original expression (A+B)(A+C) requires 3 gates.

The simplified version A+BC needs only 2 gates.

That means: cheaper, faster, smaller circuit.

Example 2: AB + AC = A(B + C)

Expand RHS: A(B + C) = AB + AC
Compare with LHS: Both same.
✅ Equation always TRUE.

Example 3: Apply De Morgan's Law

Given Expression:
¬(A+B)

This means: NOT of (A OR B)

Step 1: Apply De Morgan's Law

De Morgan's Law says:

¬(A+B) = ¬A·¬B

This flips:

  • OR to AND
  • Each variable gets a NOT
✅ Final Answer: ¬A·¬B

💡 Real-World Analogy:

Imagine a statement: "It is not true that it's raining or snowing."

This means: It's not raining AND it's not snowing.

Which is exactly what De Morgan's Law shows us.

Let's verify with a truth table:
A B A + B ¬(A + B) ¬A ¬B ¬A • ¬B
0 0 0 1 1 1 1
0 1 1 0 1 0 0
1 0 1 0 0 1 0
1 1 1 0 0 0 0

✔️ Both sides give the same result — so the identity is valid!

✅ Summary:

Original After Applying De Morgan
¬(A+B) ¬A·¬B

This simplification is especially useful when designing circuits with NAND or NOR gates.

Example 4: Associative Law (OR Operation)

Given Equation:
(A+B)+C = A+(B+C)
Why is this true?

This is the Associative Law of OR:

The grouping (parentheses) does not affect the result for OR ( + ).

So both:

(A+B)+C

and

A+(B+C)

give the same result in Boolean logic.

✅ Final Simplified Expression: A+B+C

✅ You can remove the brackets safely when only using OR.

Let's verify with a truth table:
A B C A + B (A + B) + C B + C A + (B + C) Result
0 0 0 0 0 0 0 ✅
0 0 1 0 1 1 1 ✅
0 1 0 1 1 1 1 ✅
0 1 1 1 1 1 1 ✅
1 0 0 1 1 0 1 ✅
1 0 1 1 1 1 1 ✅
1 1 0 1 1 1 1 ✅
1 1 1 1 1 1 1 ✅

✔️ Both sides are always equal, no matter the inputs.

💡 Real-World Analogy:

Suppose you're checking: "Is A, B, or C true?"

It doesn't matter if you check A and B first, then C, or B and C first, then A — the answer will be the same.

✅ Summary:

Law Type OR Form AND Form
Associative (A+B)+C = A+(B+C) (AB)C = A(BC)
Effect Grouping doesn't change result Grouping doesn't change result

Why Simplification Matters in Computers

Computers are built with logic gates.

Each gate costs transistors, power, and time.

Simplifying Boolean expressions = fewer gates.

👉 Example:

Expression 1: (A + B)(A + C) = 4 gates

Expression 2 (simplified): A + BC = only 3 gates

Millions of circuits → huge savings in CPU design.

✅ In summary:

Boolean algebra is the mathematical backbone of digital circuits.

Laws & theorems make it possible to simplify logic.

Simplification = faster, cheaper, more efficient computers.