What is Boolean algebra?

Boolean algebra is a mathematical system used to analyze logic values, where variables represent binary states (1 for true, 0 for false).

What are the main Boolean operators?

The primary operators are AND (multiplication, $\cdot$), OR (addition, $+$), and NOT (negation, $A'$ or $\bar{A}$). Other operators include XOR, NAND, and NOR.

What is a truth table?

A truth table is a tabular representation of a Boolean function showing the output value for every possible combination of input variables.

What is a Karnaugh Map (K-map)?

A Karnaugh Map is a graphical grid representation of a Boolean function, ordered using Gray code, used to simplify logic expressions without algebraic manipulation.

Why are cells in a K-map ordered in Gray code?

Gray code ensures that only one variable changes state between adjacent cells, allowing adjacent cells with 1s to be grouped and simplified.

What is the Sum-of-Products (SOP) form?

SOP is a Boolean expression format consisting of a sum (OR) of product terms (ANDs), such as $AB + A'C$.

What is the Product-of-Sums (POS) form?

POS is a Boolean expression format consisting of a product (AND) of sum terms (ORs), such as $(A + B) \cdot (A' + C)$.

What are De Morgan's laws?

De Morgan's laws are rules for negating compound expressions: $(A+B)' = A'B'$ and $(AB)' = A' + B'$.

A minterm is a product term (AND) containing all variables in either their complemented or uncomplemented form, representing a single row in a truth table where the output is 1.

What is a don't care condition?

A don't care condition (marked as X) represents an input combination that cannot occur or whose output does not matter, allowing it to be grouped as a 1 or 0 to simplify the expression.

How do you group cells in a Karnaugh Map?

Group adjacent cells containing 1s in rectangular shapes of size 1, 2, 4, 8, or 16. Groups can wrap around columns and rows.

Why is logic minimization important in hardware design?

Minimizing logic expressions reduces the number of logic gates needed on a chip, which lowers cost, power consumption, and physical size while increasing speed.

math calculator

Boolean Algebra Simplifier & K-Map Solver

Simplify Boolean logic expressions, generate truth tables, and visualize Karnaugh Maps.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Variables Count

Results

Simplified Sum-of-Products (SOP)

A'B + AB'

Raw Minterm SumA'B + AB'

Karnaugh Map (K-Map) representation

B = 0

B = 1

A = 0

A = 1

Interactive Truth Table

Row	Input A	Input B
0	0	0
1	0	1
2	1	0
3	1	1

Step-by-Step Minimization Steps

Construct the Truth Table for a 2-variable system:

Identify all minterms (rows where output equals 1).

The raw Sum-of-Products (SOP) expression is the logical OR of these minterms:

\[F = A'B + AB'\]

After applying Boolean algebra simplification laws:

\[F_{\text{simplified}} = A'B + AB'\]

Formula

Boolean Minimization Formulas

F = \sum m_i, \quad A + A' = 1, \quad A(B + C) = AB + AC

A Karnaugh Map lists minterms using Gray code, grouping blocks of 1s in powers of 2 to simplify expressions graphically.

Step by step

How the calculation works

1Select the number of variables (2-variable or 3-variable).
2Toggle the outputs (0 or 1) in the truth table below by clicking on them.
3The calculator simplifies the Boolean logic and displays the corresponding Karnaugh Map.

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math calculator

Boolean Algebra Simplifier & K-Map Solver

Simplify Boolean logic expressions, generate truth tables, and visualize Karnaugh Maps.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Variables Count

Results

Simplified Sum-of-Products (SOP)

A'B + AB'

Raw Minterm SumA'B + AB'

Karnaugh Map (K-Map) representation

B = 0

B = 1

A = 0

A = 1

Interactive Truth Table

Row	Input A	Input B
0	0	0
1	0	1
2	1	0
3	1	1

Step-by-Step Minimization Steps

Construct the Truth Table for a 2-variable system:

Identify all minterms (rows where output equals 1).

The raw Sum-of-Products (SOP) expression is the logical OR of these minterms:

\[F = A'B + AB'\]

After applying Boolean algebra simplification laws:

\[F_{\text{simplified}} = A'B + AB'\]

Formula

Boolean Minimization Formulas

F = \sum m_i, \quad A + A' = 1, \quad A(B + C) = AB + AC

A Karnaugh Map lists minterms using Gray code, grouping blocks of 1s in powers of 2 to simplify expressions graphically.

Step by step

How the calculation works

1Select the number of variables (2-variable or 3-variable).
2Toggle the outputs (0 or 1) in the truth table below by clicking on them.
3The calculator simplifies the Boolean logic and displays the corresponding Karnaugh Map.

Related guides

Learn more about this calculation

Guide

Convert values into Binary and Hexadecimal.