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Interactive Truth Table
Detected variables
ABC
ABCA∨B~C∨A
FFFFT
FFTFF
FTFTT
FTTTF
TFFTT
TFTTT
TTFTT
TTTTT

Explore boolean expressions with no friction

A truth table is a direct way to see how a logical expression behaves across every possible combination of its input values. Instead of checking each case manually, you get the full result in a single view.

On this page, you can type one or more expressions, the calculator detects the variables automatically, and the table is generated in real time. That makes it easier to study boolean logic and to validate rules used in programming and digital circuits.

  1. Automatically detects variables and builds every possible value combination.
  2. Accepts multiple expressions in the same analysis, separated by commas.
  3. Helps you study propositional logic, programming, and digital electronics clearly.

What is a truth table?

A truth table is a chart that shows the result of a logical expression for every possible combination of its input values. Each row represents a different scenario, and each column shows how the expression behaves as variables switch between true and false.

Each new variable doubles the number of rows in the table. That is why expressions with few variables are easy to inspect, while larger ones require more attention to avoid missing any combination.

How the table is generated

The process is automatic: first you enter the expression, then the system identifies the variables and organizes every possible combination. After that, each expression is evaluated row by row to show whether the result is true or false.

  1. Type one or more boolean expressions, separated by commas, such as A∧B or ¬A∨C.
  2. The system automatically detects the variables used in each expression.
  3. The table generates every possible true/false combination for those variables.
  4. Each expression is evaluated across all rows, making the final result easy to read.

This flow makes analysis faster, reduces interpretation mistakes, and keeps the logical behavior of each expression visible.

Supported logical operators

¬Negation: flips the logical value of an expression.
∧Conjunction: true only when both sides are true.
∨Disjunction: true when at least one side is true.
→Implication: false only when the first part is true and the second is false.
↔Biconditional: true when both sides have the same logical value.
⊻Exclusive OR: true when exactly one side is true.

Number of rows

A truth table with n boolean variables always has 2^n rows. Each new variable doubles the number of possible true and false combinations.

L=2n\mathbf{L = 2^n}
LTotal number of rows generated by the truth table.
nNumber of boolean variables analyzed in the expression.

With 1 variable, there are 2 rows. With 2 variables, there are 4. With 3 variables, there are 8. This exponential growth is why more complex expressions require a more careful inspection.

Example with two variables

If you use variables A and B, the truth table will have four rows and cover every possible combination.

For the expression A ∧ B, the result is true only when both A and B are true at the same time.

  1. In the first row, A and B are both false, so A ∧ B remains false.
  2. In the second row, only one variable is true, so the result is still false.
  3. The third row has the same pattern: one input is still false.
  4. In the last row, A and B are both true, so the expression becomes true.

This pattern makes the truth table behavior obvious: it shows exactly which scenarios make the expression accept or reject a combination of values.

Practical example

If you type A∨B, the table shows that the expression is false only when both variables are false. That makes the underlying logic easy to spot at a glance.

With ¬A∨B, the result changes according to the negation of A, which is useful for testing conditions and validating behavior in algorithms and circuits.

This kind of reading is useful in propositional logic, conditional programming, digital electronics, and the construction of more reliable decision rules.

Why this matters

Truth tables are not just an academic exercise. They are used whenever we need to validate logical rules, simplify conditions, or understand the behavior of an expression before putting it into practice.

  1. They help verify whether a logical rule is correct before using it in code or circuits.
  2. They make propositional logic, boolean algebra, and digital electronics easier to study.
  3. They provide a base for simplifying expressions and analyzing complex conditions.

FAQ

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01. How many expressions can I analyze at once?

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02. Which operators does the calculator understand?

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03. What happens if I enter an expression without variables?

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04. Are the variables sorted in any way?

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05. What is this useful for in practice?

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