Boolean Algebra Simplifier

Boolean algebra is a system of logic used in digital circuits and programming, where expressions combine variables with AND, OR, and NOT operations — and simplifying them reduces complexity in real designs. Enter a Boolean expression using + for OR, · for AND, and ! for NOT into the Boolean Algebra Simplifier, then choose your output format and whether to show step-by-step simplification. You'll get the simplified expression, original vs. simplified term counts, and percentage reduction.

Use + for OR, · for AND, ! for NOT. Variables can be A-Z.

Display step-by-step simplification process

Find DNF and CNF forms

Results

Simplified Expression

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Original Terms

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Simplified Terms

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Reduction

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Results Table

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Frequently Asked Questions

What Boolean algebra laws does this calculator use?

The calculator applies commutative law, distributive law, absorption law, De Morgan's theorem, idempotent law, complement law, identity law, and double negation law to simplify expressions step-by-step.

How do I enter Boolean expressions correctly?

Use + for OR operations, · (dot) for AND operations, and ! for NOT operations. Use parentheses to group terms. Variables should be single letters A-Z. Example: (A + B) · (!C + D)

What is the difference between DNF and CNF?

DNF (Disjunctive Normal Form) expresses the Boolean function as a sum of products, while CNF (Conjunctive Normal Form) expresses it as a product of sums. Both are standard canonical forms used in Boolean algebra.

Can this calculator handle complex nested expressions?

Yes, the calculator can simplify complex Boolean expressions with multiple levels of nesting and parentheses. It processes expressions systematically using established Boolean algebra rules.

Why is Boolean expression simplification important?

Simplifying Boolean expressions reduces circuit complexity, improves performance in digital systems, makes logic easier to understand, and can reduce implementation costs in hardware design.

What operators are supported besides AND, OR, and NOT?

The basic calculator supports AND (+), OR (·), and NOT (!). Advanced versions may include XOR, NAND, and NOR operations. Check the help documentation for complete operator support.

How accurate are the simplification results?

The calculator uses proven Boolean algebra laws and produces mathematically correct simplifications. All transformations preserve the logical equivalence of the original expression while reducing complexity.