Boolean Expression Simplifier
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Boolean Expression Input
Simplified Expression
Truth Table
Gate Network
How Boolean Expression Simplification Works
Boolean algebra simplification reduces complex logical expressions into their minimal equivalent form using a set of algebraic laws. Every digital circuit, from basic gates to complex processors, operates on boolean logic — and simplification directly reduces gate count, power consumption and propagation delay in real hardware.
Sum of Products (SOP)
The standard form where AND terms are ORed together. Every boolean function can be expressed in SOP and it maps directly to a two-level AND-OR gate network.
Prime Implicants
A prime implicant is a product term that cannot be combined with another term to produce a simpler term. The Quine-McCluskey method finds all prime implicants systematically.
Essential Prime Implicants
An essential prime implicant covers at least one minterm that no other prime implicant covers. The minimum cover must include all essential prime implicants.
De Morgan's Laws and Boolean Identities
These fundamental identities allow you to transform and simplify boolean expressions. The simplifier applies these automatically.
| Identity Name | Expression | Description |
|---|---|---|
| De Morgan 1 | (AB)' = A' + B' | NAND equals OR of complements |
| De Morgan 2 | (A+B)' = A'B' | NOR equals AND of complements |
| Absorption | A + AB = A | Removes redundant product terms |
| Consensus | AB + A'C + BC = AB + A'C | Eliminates redundant consensus terms |
| Complement | A + A' = 1 | AA' = 0 | A variable OR its complement is always 1 |
| Idempotent | A + A = A | AA = A | Repeated terms collapse to a single term |
Quine-McCluskey Algorithm Step by Step
The Quine-McCluskey method is a tabular technique that systematically finds the minimum SOP form of a boolean function. Unlike K-maps it works for any number of variables and is computer-friendly.
- 1 List all minterms. Convert each row where the output is 1 to its binary representation and group by number of 1-bits.
- 2 Merge adjacent groups. Combine pairs that differ in exactly one bit position, replacing that bit with a dash (don't-care). Repeat until no more merges are possible.
- 3 Identify prime implicants. Any term that could not be merged further is a prime implicant. Collect all of them into a prime implicant chart.
- 4 Select essential prime implicants. Find every minterm covered by exactly one prime implicant — that implicant is essential. Include all essentials in the final expression.
- 5 Cover remaining minterms. For any minterms not yet covered by essentials, choose additional prime implicants greedily to minimize the total term count.
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