Absorption law
A + AB
A + AB = A.
Whenever A is 1 the output is already 1, so AB cannot add another true case.
Load absorption example →DIGITAL LOGIC / BOOLEAN ALGEBRA
Boolean Algebra Simplifier with Exact Steps and Logic Circuit Output
Reduce a Boolean expression to every minimum SOP and POS form. Follow the identities and exact combination steps, inspect equivalent NAND-only and NOR-only forms and copy implementation-ready output.
Enter a Boolean expression
Exact simplified forms
Variables
Minterms
Prime implicants
Minimum solutions
Minimal SOP solutions
Minimal POS solutions
Check my next simplification step
Enter the expression you believe is the next valid line. The checker tests every input combination and gives a counterexample if the step changes the function.
Law-by-law derivation
Follow the expression from its original form through truth-table equivalence, canonical expansion, combining, prime-implicant selection and the exact minimum result.
Exact tabulation details
Prime implicant chart
Logic circuit, universal gates and implementation output
NAND-only form
NOR-only form
Verilog
VHDL
LaTeX
Worked Boolean algebra simplification examples
Each example highlights a different law or implementation goal. Load it to inspect the exact minimum forms, named derivation steps, truth table and gate output.
De Morgan’s law
NOT(A + B)
(A + B)′ = A′B′.
Move the complement through the bracket, complement both literals and change OR to AND.
Load De Morgan example →Consensus theorem
AB + A′C + BC
AB + A′C; BC is redundant.
When BC is true, either A or A′ makes one of the first two terms true.
Load consensus example →Combining adjacent terms
AB + AB′
AB + AB′ = A(B + B′) = A.
B changes between the two terms and therefore disappears from the minimum result.
Load combination example →XOR implementation
A XOR B
Minimum SOP: A′B + AB′; minimum POS: (A + B)(A′ + B′).
The universal-gate output shows how the same function can be built using only NAND or only NOR gates.
Load XOR example →Full-adder carry
AB + AC + BC
The three two-input terms implement the majority function: output 1 when at least two inputs are 1.
The truth table, K-map link and HDL output connect the algebra to a practical carry circuit.
Load carry example →How to use the Boolean algebra simplifier
Enter an expression using variables A–H and any supported word, electronics or logic-symbol notation. Add don’t-care minterms only when those input states are genuinely unspecified. The result gives exact SOP and POS minima, combination rounds and implementation forms.
Boolean algebra laws reference
| Law | OR form | Dual AND form |
|---|---|---|
| Identity | A + 0 = A | A · 1 = A |
| Null | A + 1 = 1 | A · 0 = 0 |
| Idempotent | A + A = A | A · A = A |
| Complement | A + A′ = 1 | A · A′ = 0 |
| Absorption | A + AB = A | A(A + B) = A |
| De Morgan | (A + B)′ = A′B′ | (AB)′ = A′ + B′ |
De Morgan’s laws
When a complement crosses parentheses, AND changes to OR or OR changes to AND, and every literal is complemented. For example, (A + B)′ = A′B′. Complementing only one literal or forgetting to change the operator produces a different function.
Absorption and the consensus theorem
Absorption removes a term already covered by another: A + AB = A. The consensus theorem removes a redundant third term: AB + A′C + BC = AB + A′C. The term BC cannot create a new 1-row because either A or A′ already covers it.
Worked simplification example
Start with A′BC + ABC + AB′C. Combine the first two terms because only A changes: A′BC + ABC = BC. Combine ABC and AB′C because only B changes: ABC + AB′C = AC. The minimum result is therefore BC + AC.
Why algebra may stop before the minimum
A valid algebraic rewrite depends on recognizing the right identity and arranging terms helpfully. The exact engine instead evaluates the function, generates prime implicants and solves the cover problem. Use the Karnaugh map solver when you want to inspect the same combinations visually.
Duality in Boolean algebra
Swap OR with AND and swap 0 with 1 to obtain the dual of an identity. Variables and complements remain unchanged. This is why many laws appear in matched OR and AND forms.
Why simplify a logic function?
Fewer literals can reduce gate inputs, component count, wiring, power and propagation delay. The best implementation still depends on the available gates and fan-in limits, so compare SOP, POS, NAND-only and NOR-only forms rather than assuming one representation is always cheapest.
Common Boolean algebra mistakes
- Treating + as arithmetic addition instead of OR.
- Applying De Morgan’s law to only part of a grouped expression.
- Dropping a complement when combining terms.
- Ignoring NOT-before-AND-before-OR precedence.
- Calling an expression minimum because it is shorter than the original.
Check the result by truth table
A rewrite is valid only if every defined input row matches. Open the truth table generator to compare the original and simplified expressions and obtain a counterexample if they differ.
Boolean algebra simplifier frequently asked questions
What are the basic laws of Boolean algebra?
The most frequently used laws include identity, null, idempotent, complement, involution, absorption, distributive, De Morgan and consensus.
How do I apply De Morgan’s laws?
Complement every literal inside the group and swap AND with OR or OR with AND.
What is the absorption law?
Absorption removes a term already included by another, such as A + AB = A.
What is the consensus theorem?
It identifies a redundant term: AB + A′C + BC = AB + A′C.
How do I know an expression is fully simplified?
You need to show that no equivalent cover uses fewer terms and then fewer literals. The exact minimization stage performs that check.
Can two different minimal answers both be correct?
Yes. Different prime-implicant covers can tie for the same minimum term and literal counts.
Does simplification change the truth table?
No. The simplified expression must match the original for every defined input.
Are don’t-care terms required in the result?
No. Use them only when they enable a simpler valid cover.
What is the dual of a Boolean expression?
Swap AND with OR and swap 0 with 1 while leaving variables and complements unchanged.
Why are NAND and NOR universal gates?
Either gate type can implement NOT, AND and OR, so either can build any Boolean function.