Truth table

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This page belongs to resource collections on Logic and Inquiry.

A truth table is a tabular array that illustrates the computation of a logical function, that is, a function of the form Failed to parse (Missing texvc executable; please see math/README to configure.):

where  is a non-negative integer and Failed to parse (Missing texvc executable; please see math/README to configure.): 
is the domain of logical values Failed to parse (Missing texvc executable; please see math/README to configure.): 
 The names of the logical values, or truth values, are commonly abbreviated in accord with the equations Failed to parse (Missing texvc executable; please see math/README to configure.): 
and Failed to parse (Missing texvc executable; please see math/README to configure.): 


In many applications it is usual to represent a truth function by a boolean function, that is, a function of the form where is a non-negative integer and is the boolean domain In most applications Failed to parse (Missing texvc executable; please see math/README to configure.):

is represented by  and Failed to parse (Missing texvc executable; please see math/README to configure.): 
is represented by  but the opposite representation is also possible, depending on the overall representation of truth functions as boolean functions.  The remainder of this article assumes the usual representation, taking the equations Failed to parse (Missing texvc executable; please see math/README to configure.): 
and Failed to parse (Missing texvc executable; please see math/README to configure.): 
for granted.

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 Failed to parse (Missing texvc executable; please see math/README to configure.):

also written Failed to parse (Missing texvc executable; please see math/README to configure.): 
appears below:


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.):


The negation of a proposition may be found notated in various ways in various contexts of application, often merely for typographical convenience. Among these variants are the following:


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.): bar
Failed to parse (Missing texvc executable; please see math/README to configure.): tilde
prime
complement
bang


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 Failed to parse (Missing texvc executable; please see math/README to configure.):

also written Failed to parse (Missing texvc executable; please see math/README to configure.): 
or Failed to parse (Missing texvc executable; please see math/README to configure.): 
appears below:


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.):


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 Failed to parse (Missing texvc executable; please see math/README to configure.):

also written Failed to parse (Missing texvc executable; please see math/README to configure.): 
appears below:


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.):


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 Failed to parse (Missing texvc executable; please see math/README to configure.):

also written Failed to parse (Missing texvc executable; please see math/README to configure.): 
Failed to parse (Missing texvc executable; please see math/README to configure.): 
or Failed to parse (Missing texvc executable; please see math/README to configure.): 
appears below:


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.):


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 Failed to parse (Missing texvc executable; please see math/README to configure.):

also written  or Failed to parse (Missing texvc executable; please see math/README to configure.): 
appears below:



The following equivalents may then be deduced:

Logical implication

The logical implication relation and the material conditional function 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 Failed to parse (Missing texvc executable; please see math/README to configure.):

symbolized Failed to parse (Missing texvc executable; please see math/README to configure.): 
and the logical implication Failed to parse (Missing texvc executable; please see math/README to configure.): 
symbolized Failed to parse (Missing texvc executable; please see math/README to configure.): 
appears below:


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.):


Logical NAND

The logical NAND 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 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 Failed to parse (Missing texvc executable; please see math/README to configure.):

also written Failed to parse (Missing texvc executable; please see math/README to configure.): 
or Failed to parse (Missing texvc executable; please see math/README to configure.): 
appears below:


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.):


Logical NNOR

The logical NNOR (“Neither Nor”) 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 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 Failed to parse (Missing texvc executable; please see math/README to configure.):

also written Failed to parse (Missing texvc executable; please see math/README to configure.): 
appears below:


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.):


Translations

Syllabus

Focal nodes

Peer nodes

Logical operators

Related topics

Relational concepts

Information, Inquiry

Related articles

Document history

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.