Truth table
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A truth table is a tabular array that illustrates the computation of a boolean function, that is, a function of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{B}</math> is the boolean domain <math>\{ 0, 1 \}.\!</math>
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Logical negation[edit]
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 |
---|---|
<math>\bar{p}</math> | bar p |
<math>p'\!</math> | p prime, p complement |
<math>!p\!</math> | bang p |
Logical conjunction[edit]
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<math>\cdot</math>q) is as follows:
p | q | p ∧ q |
---|---|---|
F | F | F |
F | T | F |
T | F | F |
T | T | T |
Logical disjunction[edit]
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[edit]
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[edit]
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:
- <math>\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}</math>
Logical implication[edit]
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[edit]
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[edit]
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 |
Translations[edit]
Syllabus[edit]
Focal nodes[edit]
<p>Peer nodes[edit]
Logical operators[edit]
Related top.css[edit]
Relational concepts[edit]
Information, Inquiry[edit]
Related articles[edit]
Document history[edit]
Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.