Truth table
A truth table is a tabular array that illustrates the computation of a boolean function, that is, a function of the form \(f : \mathbb{B}^k \to \mathbb{B},\) where \(k\!\) is a non-negative integer and \(\mathbb{B}\) is the boolean domain \(\{ 0, 1 \}.\!\)
Logical negation
Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true.
The truth table of NOT p (also written as ~p or ¬p) is as follows:
| p | ¬p |
|---|---|
| F | T |
| T | F |
The logical negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:
| Notation | Vocalization |
|---|---|
| \(\bar{p}\) | bar p |
| \(p'\!\) | p prime, p complement |
| \(!p\!\) | bang p |
Logical conjunction
Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.
The truth table of p AND q (also written as p ∧ q, p & q, or p\(\cdot\)q) is as follows:
| p | q | p ∧ q |
|---|---|---|
| F | F | F |
| F | T | F |
| T | F | F |
| T | T | T |
Logical disjunction
Logical disjunction, also called logical alternation, is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false.
The truth table of p OR q (also written as p ∨ q) is as follows:
| p | q | p ∨ q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
Logical equality
Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.
The truth table of p EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:
| p | q | p = q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | T |
Exclusive disjunction
Exclusive disjunction, also known as logical inequality or symmetric difference, is an operation on two logical values, typically the values of two propositions, that produces a value of true just in case exactly one of its operands is true.
The truth table of p XOR q (also written as p + q, p ⊕ q, or p ≠ q) is as follows:
| p | q | p XOR q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | F |
The following equivalents can then be deduced:
\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ \\ & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\ \\ & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}\]
Logical implication
The logical implication and the material conditional are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if the first operand is true and the second operand is false.
The truth table associated with the material conditional if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:
| p | q | p ⇒ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
Logical NAND
The logical NAND is a logical operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are true. In other words, it produces a value of true if and only if at least one of its operands is false.
The truth table of p NAND q (also written as p | q or p ↑ q) is as follows:
| p | q | p ↑ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | T |
| T | T | F |
Logical NNOR
The logical NNOR is a logical operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are false. In other words, it produces a value of false if and only if at least one of its operands is true.
The truth table of p NNOR q (also written as p ⊥ q or p ↓ q) is as follows:
| p | q | p ↓ q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | F |
