Boolean Functions – The Digital DNA
Boolean Algebra, introduced by George Boole, is the backbone of digital logic design. Think of it as a special kind of math that deals only with two values: 0 (false) and 1 (true). Boolean expressions represent logic using variables, operators, and constants—and are key to designing circuits.
When multiple logical inputs control a digital output, we describe that control using Boolean functions.
Truth Table – The Blueprint of Logic
A Truth Table lists every possible combination of inputs for a logic function and shows what output those inputs will generate.
📌 Formula:
Total combinations = 2ⁿ
where n is the number of input variables.
Example:
2-Variable Truth Table:
| A | B | (A + B) | (A·B) | (~A) | (~B) |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 | 0 | 0 |
Methods to Solve Boolean Functions
1. Sum of Products (SOP)
SOP simplifies logic by combining input conditions (minterms) where the output is 1.
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Each minterm is an AND of variables.
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All such terms are added (OR-ed) together.
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Represented as: Σm
Example:
From the truth table below, select rows where output Y = 1.
| A | B | C | Y |
|---|---|---|---|
| 0 | 0 | 0 | 1 → m₀ |
| 0 | 1 | 0 | 1 → m₂ |
| 0 | 1 | 1 | 1 → m₃ |
| 1 | 1 | 0 | 1 → m₆ |
| 1 | 1 | 1 | 1 → m₇ |
POS focuses on where the output is 0 (maxterms). It's the opposite of SOP.
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Each maxterm is an OR of variables.
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All such terms are ANDed together.
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Represented as: ΠM
Example:
Pick the rows where Y = 0.
| A | B | C | Y |
|---|---|---|---|
| 0 | 0 | 0 | 0 → M₀ |
| 0 | 1 | 1 | 0 → M₃ |
| 1 | 0 | 0 | 0 → M₄ |
| 1 | 1 | 1 | 0 → M₇ |
Karnaugh Map (K-Map) – Puzzle to Logic:
A K-Map is a visual technique to minimize Boolean expressions. It helps group 1s in blocks of 2ⁿ (1, 2, 4, 8,...) to reduce complexity in SOP or POS expressions.
Example:
Function: F(A, B, C) = Σm(0, 1, 2, 4, 7)
| A | B | C | F |
|---|---|---|---|
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 |
Group adjacent 1s and simplify:
NAND Implementation – Universal Gate Magic:
The NAND Gate is powerful—it can recreate all other logic gates. It outputs 0 only when all inputs are 1.
Truth Table:
|
|---|
Boolean Function with NAND Gates:
Take:
F(A, B, C) = ABC + A'B' + B'C' + A'C'
To implement with only NAND:
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Convert each term into NAND form using De Morgan’s laws.
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Replace all basic gates with NAND equivalents.
Each AND becomes:
A·B = ¬(¬A + ¬B)
Each OR becomes:
And each NOT becomes:
¬A = A NAND A
Conclusion:
Boolean functions are the heart of digital electronics. Whether it's simplifying logic with SOP and POS, or optimizing circuits with K-Maps and NAND gates—understanding these core tools helps you build smarter, faster, and more efficient digital systems.
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