Formula & Calculator
De Morgan's Theorem
Provides the relationship between AND, OR, and NOT operations for simplifying Boolean expressions.
Interpretation
De Morgan's theorem: the complement of an AND is the OR of complements, and the complement of an OR is the AND of complements.
(A·B)' = A' + B' and (A+B)' = A' · B'.
Example: (A·B)' = A' + B' is used extensively to simplify logic circuits.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| A | Input A | dimensionless |
| B | Input B | dimensionless |
| Y | Output (A·B)' | dimensionless |
| Z | Output A' + B' | dimensionless |
What it means
De Morgan’s theorem provides rules for simplifying Boolean expressions. It states that the complement of an AND is the OR of complements: (A·B)' = A' + B', and the complement of an OR is the AND of complements: (A+B)' = A' · B'. These transformations are essential for converting between positive and negative logic, and for simplifying complex logic circuits. They allow the use of different gate types (NAND vs NOR) and are heavily used in digital logic design. De Morgan’s theorem also helps in minimising logic functions using Boolean algebra. Example: To implement a function F = (A·B)' using only NOR gates, De Morgan gives F = A' + B', which can be built with two inverters and a NOR gate, or directly with a NAND gate. This theorem is fundamental for digital electronics.
Worked example
De Morgan's Theorem – Practical Example
Real‑World| Parameter | Value |
|---|---|
| Inputs | A, B |
| Original | (A·B)' |
| Theorem | (A·B)' = A' + B' |
Common mistakes
Watch unit consistency and the assumptions behind the formula; misapplying it outside its valid conditions is the most frequent error.Applications
De Morgan's theorem states that the complement of an AND is the OR of complements, and vice versa: (A·B)' = A' + B' and (A+B)' = A'·B'. This theorem is widely used to simplify Boolean expressions and to convert between gate types. Engineers use it to reduce the number of gates, to minimise circuit area, and to improve speed. By applying De Morgan's laws, they can transform complex logic into more implementable forms. This is a core tool in digital logic design and is essential for combinational logic optimisation.
- Logic simplification and gate‑level minimisation
- Conversion between NAND/NOR and AND/OR implementations
- Design of efficient digital circuits
- Boolean algebra and Karnaugh maps
- Educational foundation of digital logic design
Frequently Asked Questions
De Morgan's theorem provides the relationship between AND, OR, and NOT. There are two forms: (A·B)' = A' + B' and (A+B)' = A'·B'.
It allows converting between AND/OR forms and helps in simplifying Boolean expressions. For example, it is used to implement AND gates using NOR gates.
The two forms are duals: complement of the product is the sum of complements; complement of the sum is the product of complements.
Yes: (A·B·C)' = A' + B' + C' and (A+B+C)' = A'·B'·C'.
It is used to convert circuits to use only NAND or only NOR gates, which are often easier to implement.
Common errors: 1) forgetting to complement each variable, 2) applying to only part of an expression, 3) using it incorrectly for XOR/XNOR, 4) missing the parentheses.
De Morgan's theorem is a special case of duality. The dual of an expression is obtained by swapping AND/OR and 0/1.
Used in logic synthesis, digital circuit optimization, and error checking in Boolean algebra.