**Math Calculators** ▶ Truth Table Calculator

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An online truth table calculator will provide the truth table values for the given propositional logic formulas. The propositional logic statements can only be true or false. Many statements can be combined with logical connections to form new statements. The truth table solver generates all combinations of true and false statements and calculates the corresponding truth content of the logical expression.

The truth table is a tabular view of all combinations of values for the inputs and their corresponding outputs. It is a mathematical table that shows all possible results that may be occur from all possible scenarios. It is used for logic tasks such as logic algebra and electronic circuits.

The truth table shows the result of the logical expression, each involved variable has a separate column, and the corresponding result has a column. All variants of the input data and their parameters are listed on the left; the output is usually placed in the last column on the right.

A proposition is a set of declarative statements with a truth value of “true” or a truth value of “false”. Propositional expressions are composed of connectives and propositional variables. We use capital letters to represent the propositional variables (A, B). The connectives connect the propositional variables.

Here are some examples of propositions-

“12 + 9 = 3-2”, returns true “FALSE”

“Man is mortal”, actually returns “TRUE”

The following is not a proposition-

“X is less than 2”.

This is because unless we give X as a specific value, we cannot judge whether this statement is right or wrong.

However, an online One’s Complement Calculator can easily implement a logic circuit with only NOT gate for every bit of Binary number input.

In propositional logic truth table calculator uses the different connectives which are −

- OR (∨)
- AND (∧)
- Negation/ NOT (¬)
- Implication / if-then (→)
- If and only if (⇔)
- Absurdity (#)
- Sheffer Stroke (|)

The OR operation of two propositions such as P and Q (written as P∨Q) is true if at least any of the propositional variable P or Q is true.

The truth table examples are as follows −

P |
Q |
P ∨ Q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

The AND operation of two propositions P and Q (written as P∧Q) is true if both the propositional variable P and Q is true.

The truth table is as follows −

P |
Q |
P ∧ Q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

The negation of a proposition P (written as ~P) is true when the value of “P” is false and negation is false when “P” value is true.

The negation for the P variable as:

P |
~ P |

T | F |

F | T |

An implication P→ Q is the proposition “if P, then Q”. It is false if P is true and Q is false. The rest cases are true.

The implication truth table is as follows −

Q |
Q |
P → Q |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

Therefore, this is the truth table for implication with two different variables.

P⇔Q is bi-conditional logical connective which is true when P and Q are same. For example, both are true or both are false.

The truth table is as follows −

P |
Q |
P ⇔ Q |

T | T | T |

T | F | F |

F | T | F |

F | F | T |

A Tautology is an equation, which is always true for each value of its variables.

**Example:**

Prove [(P → Q) ∧ P] → Q is a tautology

The truth table maker prove the tautology for P and Q variables by the following table:

P |
Q |
P → Q |
(P → Q) ∧ P |
[( P → Q ) ∧ P] → Q |

T | T | T | T | T |

T | F | F | F | T |

F | T | T | F | T |

F | F | T | F | T |

The value of [(P → Q) ∧ P] → Q from truth table generator is true. Therefore, it is a tautology.

A Contradiction is an equation, which is always false for each value of its propositional values.

**Example: **

**Prove (P ****∨ Q) ****∧ [(~P) ****∧ (~Q)] is a contradiction.**

**Solution:**

The truth table calculator display and use the following table for the contradiction −

P |
Q |
P ∨ Q |
~ P |
~ Q |
(~ P) ∧ (~ Q) |
(P ∨ Q) ∧ [( ~ P) ∧ (~ Q)] |

T | T | T | F | F | F | F |

T | F | T | F | T | F | F |

F | T | T | T | F | F | F |

F | F | F | T | T | T | F |

If p then q truth table have false values, then it is a contradiction. However, an Online Two’s Complement Calculator allows you to calculate 2’s complement of the given decimal, binary or hexadecimal number.

A Contingency is an equation, which has both some false and some true values for every value of its propositional variables.

**Example: **

**Prove (P ∨ Q) ∧ (~P) a contingency**

**Solution:**

The truth table generator displays the contingency truth table for P, Q, and ~P:

P |
Q |
P ∨ Q |
~ P |
(P ∨ Q) ∧ (~ P) |

T | T | T | F | F |

T | F | T | F | F |

F | T | T | T | T |

F | F | F | T | F |

As we can see every value of truth tables with 3 variables have both true or false outcome, it is a contingency.

Two statements A and B are logically equivalent if any of the following two conditions hold –

- The bi-conditional statement A⇔B is a tautology.
- The truth tables of every statement have the same truth variables.

**Example:**

Prove ~(P ∨ Q) and [(~P) ∧ (~Q)] are equivalent

**Solution: **

The truth tables calculator perform testing by matching truth table method

P |
Q |
P ∨ Q |
¬ (P ∨ Q) |
¬ P |
¬ Q |
[(¬ P) ∧ (¬ Q)] |

T | T | T | F | F | F | F |

T | F | T | F | F | T | F |

F | T | T | F | T | F | F |

F | F | F | T | T | T | T |

Here, we can see the truth values of ~(P ∨ Q) and [(~P) ∧ (~Q)] are same, hence all the statements are equivalent.

**Testing by Bi-conditionality method **

P |
Q |
~ (P ∨ Q ) |
[(~ P) ∧ (~ Q)] |
[~ (P ∨ Q)] ⇔ [(~ P ) ∧ (~ Q)] |

T | T | F | F | T |

T | F | F | F | T |

F | T | F | F | T |

F | F | T | T | T |

As [~(P ∨ Q)] ⇔ [(~P) ∧ (~Q)] is a tautology, the statements are equivalent.

An online truth table generator provides the detailed truth table by following steps:

- First, enter a propositional logic equation with symbols.
- Hit the calculate button for results.

- The truth table calculator construct a truth table for 4 variables of the given expression.

A table of logical expressions used to express the functions of logical elements, usually called a boolean truth table. The gate truth table shows every possible input combination of the gate or circuit, and the result output depends on the combination of these inputs.

There are 4 unary operations:

- Always true
- Never true, unary falsum
- Unary Identity
- Unary negation

Use this online truth table calculator to create the multivariate propositional logic truth tables. Propositional logic deals with statements that can be truth values, “true” and “false”. The purpose is to analyze these statements individually or collectively.

From the source of Wikipedia: Unary operations, Logical true, Logical false, Logical identity, Logical negation, Binary operations, Logical conjunction (AND), Logical disjunction (OR), Logical implication.

From the source of Tutorial Points: Prepositional Logic, Connectives, Tautologies, Contradictions, Contingency, Propositional Equivalences, Inverse, Converse, and Contra-positive.