Truth Tables

Truth Tables | Proofs | Logic Calculator | Boolean Algebra | Learn | Logic

The application generates truth tables for propositional calculus and contains proofs.

You will find proof of many logical expressions here. If you are a student or simply fascinated by logic, this application is definitely for you.

You can generate any truth table you want.

What's New in the Latest Version 1.24

Last updated on Jul 5, 2024

The application generates truth tables and shows all the steps needed to solve the problem. Here you will find proofs of propositional calculus.

Truth tables are a fundamental tool in logic and computer science. They provide a systematic way to evaluate the truth value of compound propositions, which are propositions formed by combining simpler propositions using logical connectives.

Logical Connectives

The most common logical connectives are:

* Conjunction (∧): True only if both propositions are true.

* Disjunction (∨): True if either or both propositions are true.

* Negation (¬): True if the proposition is false, and false if the proposition is true.

* Implication (→): True if the first proposition is false or the second proposition is true.

* Equivalence (↔): True if both propositions have the same truth value.

Constructing Truth Tables

To construct a truth table, list all possible combinations of truth values for the constituent propositions. Then, evaluate the compound proposition for each combination using the rules for the logical connectives.

Example

Consider the compound proposition (P ∧ Q) → R. The truth table is as follows:

| P | Q | R | (P ∧ Q) → R |

|---|---|---|---|

| True | True | True | True |

| True | True | False | False |

| True | False | True | True |

| True | False | False | True |

| False | True | True | True |

| False | True | False | True |

| False | False | True | True |

| False | False | False | True |

As we can see from the truth table, the compound proposition is true for all combinations of truth values, except when P and Q are both true and R is false.

Applications of Truth Tables

Truth tables have numerous applications in logic and computer science, including:

* Simplifying compound propositions: Truth tables can be used to identify tautologies (always true) and contradictions (always false).

* Verifying logical arguments: Truth tables can be used to determine the validity of logical arguments by showing whether the conclusion follows from the premises.

* Designing digital circuits: Truth tables are used to design and analyze digital circuits, which are used in computers and other electronic devices.