Truth table

From semanticweb.org.edu
Jump to: navigation, search

This page belongs to resource collections on Logic and Inquiry.

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>

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:


Logical Negation
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:


Variant Notations
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:


Logical Conjunction
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:


Logical Disjunction
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:


Logical Equality
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:


Exclusive Disjunction
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:


Logical Implication
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:


Logical NAND
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:


Logical NNOR
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.