Expert in Digital Logic and Theoretical Computing.
Welcome to the **Boolean Logic Calculator**, your essential tool for generating comprehensive truth tables and verifying the results of complex digital logic expressions involving variables A, B, and C. Enter your expression below using standard notation.
Boolean Logic Expression Solver
See steps below for the full truth table.
Boolean Logic Calculator Laws
De Morgan’s Laws:
NOT (A AND B) = (NOT A) OR (NOT B)
NOT (A OR B) = (NOT A) AND (NOT B)
Distributive Law:
A AND (B OR C) = (A AND B) OR (A AND C)
Formula Source: Wikipedia: Boolean Algebra, TutorialsPoint: Logical Laws
Variables Explained
This solver is designed for expressions using up to three variables:
- A, B, C: These represent the input binary variables in your digital circuit or proposition. They can only hold a value of True (1) or False (0).
- Operators: Use `AND`, `OR`, and `NOT` (or `&&`, `||`, `!`) to define the relationship between the variables.
- Parentheses: Use `( )` to explicitly control the order of operation.
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What is Boolean Logic?
Boolean logic, or Boolean algebra, is a branch of algebra that deals with logical operations and binary variables. It was invented by George Boole in the mid-19th century and forms the fundamental basis of all modern digital electronics and computer programming. Unlike elementary algebra, Boolean variables can only have two possible values: True (1) or False (0).
The key operations are Conjunction (AND), Disjunction (OR), and Negation (NOT). This system allows complex relationships and conditions to be simplified and modeled mathematically, which is essential for designing efficient microprocessors and software flow control.
Every decision a computer makes, from opening a file to executing a complex algorithm, is ultimately a series of Boolean comparisons and operations carried out at the hardware level.
How to Calculate Boolean Logic (Example)
- Define the Expression: Start with an expression, for example: $X = (A \text{ OR } B) \text{ AND } (\text{NOT } A)$.
- Identify Variables: The variables are A and B. This requires $2^2 = 4$ rows in the truth table.
- List All Inputs: Create columns for A and B, listing all combinations: (0, 0), (0, 1), (1, 0), (1, 1).
- Evaluate Inner Operations: Create intermediate columns. First, calculate the Negation: $Y = \text{NOT } A$.
- Evaluate Parentheses: Calculate the first term: $Z = (A \text{ OR } B)$.
- Determine Final Output: Use the results from steps 4 and 5 to calculate the final expression: $X = Z \text{ AND } Y$. The final column for X will show the truth value of the entire expression for every possible input.
Frequently Asked Questions (FAQ)
Is this calculator only for A, B, and C?
This tool is optimized for expressions with up to three variables (A, B, C) to keep the truth table readable (8 rows). You can use fewer variables (A and B), but the system will assume C is part of the environment.
What is the difference between an OR gate and an XOR gate?
The standard OR gate outputs TRUE if *at least one* input is TRUE. The XOR (Exclusive OR) gate outputs TRUE only if the inputs are *different* (one is TRUE and the other is FALSE). The standard OR is supported, but XOR is not explicitly included in this simplified solver.
What does it mean if my expression is a Tautology?
A tautology is a Boolean expression that is always TRUE, regardless of the values of its variables. For example, $A \text{ OR } (\text{NOT } A)$ is a tautology. If the final result column is all TRUEs, the expression is a tautology.
Can I use symbols like && and || instead of words?
Yes, the calculator supports both keywords (`AND`, `OR`, `NOT`) and the common programming symbols (`&&`, `||`, `!`) for convenience. The system automatically converts the keywords to symbols before evaluation.