Calculation Result
Step-by-Step Solution
Merged Result Report
Calculator Options
Calculation History
Formula Used
De Morgan’s theorem distributes a negation across grouped terms. The connecting operator changes during the transformation. Every grouped term also receives its own negation.
How to Use the Calculator
Select Boolean, set, comparison, or practice mode first. Enter the required expression or set values carefully. Press Calculate to generate the complete result.
Use AND, OR, NOT, symbols, or common programming operators. Parentheses may be nested inside larger expressions. The calculator validates syntax before evaluating any rows.
Review the theorem, transformation, truth table, and detailed steps. Use the export buttons for saved documentation. Reset the form whenever starting another calculation.
Worked Examples
| Type | Original | Transformed | Result |
|---|---|---|---|
| Boolean AND | ¬(A ∧ B) | ¬A ∨ ¬B | Equivalent |
| Boolean OR | ¬(A ∨ B) | ¬A ∧ ¬B | Equivalent |
| Three variables | ¬(A ∨ B ∨ C) | ¬A ∧ ¬B ∧ ¬C | Equivalent |
| Set union | (A ∪ B)ᶜ | Aᶜ ∩ Bᶜ | Equal sets |
Logic-Gate Interpretation
A NAND gate equals an OR gate with inverted inputs. A NOR gate equals an AND gate with inverted inputs. These identities support practical digital circuit conversions.
Bubble pushing is a common visual design technique. Moving an inversion bubble changes the associated gate type. De Morgan’s theorem proves that conversion remains logically equivalent.
Common Mistakes
Do not distribute negation without changing the operator. Negate every term inside the affected parentheses. Keep unrelated terms outside the transformation unchanged.
Set complements require a clearly defined universal set. Missing universal values create incomplete complement results. Duplicate entries should not change any mathematical set.
Frequently Asked Questions
1. What is De Morgan’s theorem?
It describes how negation changes grouped AND and OR operations. The operator switches while each term becomes negated. Equivalent results follow for every possible input combination.
2. Which Boolean operators are supported?
The calculator supports NOT, AND, OR, and parentheses. It accepts mathematical symbols and common programming forms. Words such as NOT and AND also work.
3. Can it handle nested expressions?
Yes, nested parentheses are parsed recursively. Each applicable negation is transformed in sequence. The final result removes unnecessary double negations.
4. How does the truth table verify equivalence?
Every possible variable assignment is generated. Both expressions are evaluated for each assignment. Matching result columns prove logical equivalence.
5. What is required for set complements?
A universal set must contain all relevant elements. Each complement is calculated relative to that set. Elements outside the universal set are rejected.
6. Can three or four sets be used?
Yes, select any two to four available sets. The theorem extends across every selected set. The calculator displays both equal set expressions.
7. Does the calculator simplify double negation?
Yes, two consecutive negations cancel automatically. The step list records that simplification. Remaining negations stay attached to their proper terms.
8. Can I compare unrelated expressions?
Yes, comparison mode accepts two valid expressions. A combined truth table checks every input row. The result reports equivalent or not equivalent.
9. Can results be exported?
Yes, copy, CSV, PDF, and print controls are included. CSV contains truth-table rows when available. PDF includes the merged calculation report.