📚 Lesson: Boolean Algebra & Core Minimization Theorems

1️⃣ Introduction

Boolean minimization is the process of simplifying Boolean expressions without changing their logical function.
Why minimize?

• ✅ Fewer logic gates → lower cost
• ✅ Shorter propagation delay → faster circuits
• ✅ Less power consumption

2️⃣ Boolean Algebra Refresher

• Identities (AND, OR, XOR, NOT)
• Properties (Commutative, Associative, Distributive)
• De Morgan’s Theorems

2.1 Boolean Identities

2.2 Boolean Properties

• Commutative: A + B = B + A, AB = BA
• Associative: (A + B) + C = A + (B + C)
• Distributive: A(B + C) = AB + AC

2.3 De Morgan’s Theorems

1. (A·B)’ = A’ + B’
2. (A + B)’ = A’·B’

💡 Tip: “Break the bar, change the operator.”

Truth Table Proof for (A·B)’ = A’ + B’:

3️⃣ Core Minimization Theorems

3.1 Absorption Theorem

Law: A + AB = A
Proof: A(1 + B) = A(1) = A
Example: X + XY = X

3.1.1 Layered Absorption

Expression: F = A + AB + ABC Step-by-step:

• A + AB → A (Absorption) Result: F = A + ABC
• A + ABC → A (Absorption again) Final: F = A 💡 Even though ABC looks “different,” it’s still absorbed by A because A is already true in all cases where ABC is true.

3.1.2 Absorption Hidden in Grouped Terms

Expression: $F = A + (A \cdot B) + (A \cdot B \cdot C’) + D$ Step-by-step:

• A + AB → A $Result: F = A + (A \cdot B \cdot C’) + D$
• A + ABC’ → A Result: F = A + D

3.1.3 Absorption with Mixed Variables

Expression: F = XY + X + XZ’ Step-by-step:

• X + XY → X Result: F = X + XZ’
• X + XZ’ → X Final: F = X

3.1.4 Absorption After Factoring

Sometimes you can’t see absorption until you factor. Expression: F = AB + A + AC Step-by-step:

• A + AB → A Result: F = A + AC
• A + AC → A Final: F = A

3.1.5 Absorption in Disguised Form

Expression: F = (P + Q) + (P + Q)R Step-by-step:

• Recognize A + AB form with A = (P + Q) and B = R
• Apply absorption: $(P + Q) + (P + Q)R \; \to \; (P + Q)$

3.1.6 Absorption Across Multiple Levels

Expression: F = M + MN + MNP + MN’Q Step-by-step:

• M + MN → M Result: F = M + MNP + MN’Q
• M + MNP → M Result: F = M + MN’Q
• M + MN’Q → M Final: F = M

3.1.7 Absorption in Consensus-like Situations

Expression: F = AB + A’B + AB’ Here, absorption doesn’t directly apply to all terms, but if you factor:

• Group: AB + AB’ = A(B + B’) = A(1) = A
• Now: F = A + A’B
• Apply absorption: A + A’B → A + B (Simplification Theorem, related to absorption)

3.2 Consensus (Optional Product) Theorem

Law: AB + A’C + BC = AB + A’C
Meaning: The term BC is redundant.
Example: XY + X’Z + YZ = XY + X’Z

3.2.1 Algebraic Proof

• We start with:

  AB + A’C + BC

• Step 1: Factor BC with (1):

  = AB + A’C + BC(1)

• Step 2: Replace 1 with A + A’:

  = AB + A’C + BC(A + A’)

• Step 3: Distribute:

  = AB + A’C + ABC + A’BC

• Step 4: Notice:

  • AB + ABC = AB (Absorption)
  • A’C + A’BC = A’C (Absorption)

So:

= AB + A'C

Q.E.D.

3.2.2 Truth Table Verification

The last two columns are identical → theorem holds.

3.2.3 More Complex Examples

Example 1 – Direct Application

F = XY + X’Z + YZ

• Consensus term: YZ
• Remove it: F = XY + X’Z

Example 2 – Variables as Sub-Expressions

Let:

• P = (A + B)
• Q = C’
• R = D Expression: F = PQ + P’R + QR
• Consensus term: QR
• Remove it: F = PQ + P’R

Example 3 – Hidden Consensus After Factoring

F = AB + A’C + BC + A’B’C

• First, notice AB + A’C + BC already fits the theorem → remove BC: F = AB + A’C + A’B’C
• Now A’C + A’B’C → A’C (Absorption) Final: F = AB + A’C

Example 4 – Consensus in Multi-Level Logic

$F = (M \cdot N) + (M’ \cdot P) + (N \cdot P)$

• $Consensus term: N \cdot P$
• Remove it: F = MN + M’P

Example 5 – Consensus with Inverted Variables

F = A’B + AC + BC

• Consensus term: BC
• Remove it: F = A’B + AC

4️⃣ Practice

Simplify: A + AB + A’B

• Step 1: Apply Absorption → A + A’B
• Step 2: Apply Simplification → A + B

5️⃣ Summary

• Boolean identities are the building blocks.
• Absorption and Consensus are key for algebraic minimization.
• De Morgan’s Theorems are essential for gate-level transformations.