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 f : \mathbb{B}^k \to \mathbb{B}, where k\! is a non-negative integer and \mathbb{B} is the boolean domain \{ 0, 1 \}.\!

Contents

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:


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
\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\cdotq) is as follows:


Logical Conjunction
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:


Logical Disjunction
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:


Logical Equality
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:


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:

\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:


Logical Implication
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:


Logical NAND
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:


Logical NNOR
p q p ↓ q
F F T
F T F
T F F
T T F


Translations

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<p>

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Logical operators

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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.

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