Truth table calculator
Table of Contents
A truth table is a mathematical table used to determine the output of a logic function based on all possible combinations of inputs. Each row represents a possible state of the input variables, with the corresponding output value. Truth tables are crucial in designing and understanding digital circuits, Boolean algebra, and logical expressions.
1. Implemented logical operations
1.1. Equivalent ( ⇔ , 1 )
The equivalent operation, also known as logical biconditional, is true if and only if both inputs are the same. In other words, it asserts that both propositions are either both true or both false. It is often represented by the symbol ⇔.
Truth Table:
| A | B | A ⇔ B |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
1.2. Implies ( ⇒ , 2 )
An implication asserts that if the first proposition is true, the second must be true as well. If the first is false, the implication holds regardless of the second proposition's value.
Truth table:
| A | B | A ⇒ B |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
1.3. OR ( ∨ , 3 )
The OR operation, also known as logical disjunction, is true if at least one of the inputs is true. It asserts that if either proposition is true, the entire expression is true.
Truth table:
| A | B | A ∨ B |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
1.4. AND ( ∧ , 4 )
The AND operation, also known as logical conjunction, is true if and only if both inputs are true.
Truth table:
| A | B | A ∧ B |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
1.5. NOT ( ¬ , 5 )
The NOT operation, also known as logical negation, inverts the value of the input. If the input is true, the output is false, and vice versa.
Truth Table:
| A | ¬A |
|---|---|
| T | F |
| F | T |
2. Examples
2.1. Example: (A ∧ B) ∨ ¬C
| A | B | C | (A ∧ B) ∨ ¬C |
|---|---|---|---|
| T | T | T | T |
| T | T | F | T |
| T | F | T | F |
| T | F | F | T |
| F | T | T | F |
| F | T | F | T |
| F | F | T | F |
| F | F | F | T |
2.2. Example: A ⇒ (B ∨ ¬C)
| A | B | C | A ⇒ (B ∨ ¬C) |
|---|---|---|---|
| T | T | T | T |
| T | T | F | T |
| T | F | T | F |
| T | F | F | T |
| F | T | T | T |
| F | T | F | T |
| F | F | T | T |
| F | F | F | T |
2.3. Example: (A ⇔ B) ∧ C
Truth Table:
| A | B | C | (A ⇔ B) ∧ C |
|---|---|---|---|
| T | T | T | T |
| T | T | F | F |
| T | F | T | F |
| T | F | F | F |
| F | T | T | F |
| F | T | F | F |
| F | F | T | T |
| F | F | F | F |