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A truth table is a tabular array that illustrates the computation of a logical function, that is, a function of the form where is a non-negative integer and is the domain of logical values The names of the logical values, or truth values, are commonly abbreviated in accord with the equations and

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 is represented by and 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 and for granted.

Logical negationEdit

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   also written   appears below:


 
   
   
   


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:


 
   
    bar
    tilde
    prime
  complement
  bang  


Logical conjunctionEdit

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   also written   or   appears below:


 
     
     
     
     
     


Logical disjunctionEdit

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   also written   appears below:


 
     
     
     
     
     


Logical equalityEdit

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   also written     or   appears below:


 
     
     
     
     
     


Exclusive disjunctionEdit

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   also written   or   appears below:


 
     
     
     
     
     


The following equivalents may then be deduced:

 

Logical implicationEdit

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   symbolized   and the logical implication   symbolized   appears below:


 
     
     
     
     
     


Logical NANDEdit

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   also written   or   appears below:


 
     
     
     
     
     


Logical NNOREdit

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   also written   appears below:


 
     
     
     
     
     


TranslationsEdit

SyllabusEdit

Document historyEdit

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.