Definition:Truth Table

Definition
A truth table is a tabular array that represents 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
 * $\mathbb B$ is a boolean domain, usually $\left\{{0, 1}\right\}$ or $\left\{{T, F}\right\}$.

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
 * a value of false when its operand is true.

The truth table of $\mathsf{NOT}\ p$ (also written $\neg p$ or $\sim p\!$) is as follows:


 * $\begin{array}{|c||c|} \hline

p & \neg p \\ \hline F & T \\ T & F \\ \hline \end{array}$

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 its operands are true.

The truth table of $p\ \mathsf{AND}\ q$ (also written $p \land q$, $p \ \And \ q$ or $p \cdot q$) is as follows:


 * $\begin{array}{|cc||c|c|} \hline

p & q & p \land q \\ \hline F & F & F \\ F & T & F \\ T & F & F \\ T & T & T \\ \hline \end{array}$

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 its operands are false.

The truth table of $p\ \mathsf{OR}\ q$ (also written $p \lor q$) is as follows:


 * $\begin{array}{|cc||c|c|} \hline

p & q & p \lor q \\ \hline F & F & F \\ F & T & T \\ T & F & T \\ T & T & T \\ \hline \end{array}$

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 $p\ \mathsf{EQ}\ q$ (also written $p = q$, $p \iff q$ or $p \equiv q$) is as follows:


 * $\begin{array}{|cc||c|c|} \hline

p & q & p \iff q \\ \hline F & F & T \\ F & T & F \\ T & F & F \\ T & T & T \\ \hline \end{array}$

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 $p\ \mathsf{XOR}\ q$ (also written $p + q$, $p \oplus q$ or $p \neq q$) is as follows:


 * $\begin{array}{|cc||c|c|} \hline

p & q & p \oplus q \\ \hline F & F & F \\ F & T & T \\ T & F & T \\ T & T & F \\ \hline \end{array}$

The following equivalents can then be deduced:

Conditional
The conditional is associated with an operation on two logical values 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 as $p \implies q$) is as follows:


 * $\begin{array}{|cc||c|c|} \hline

p & q & p \implies q \\ \hline F & F & T \\ F & T & T \\ T & F & F \\ T & T & T \\ \hline \end{array}$

Logical NAND
The logical NAND is a logical operation on two logical values, typically the values of two propositions, that produces a value of true if and only if at least one of its operands is false. In other words, it produces a value of false if and only if both of its operands are true.

The truth table of $p\ \mathsf{NAND}\ q$ (also written $p~|~q$, $p \uparrow q$ or $p \bar \curlywedge q$) is as follows:


 * $\begin{array}{|cc||c|c|} \hline

p & q & p \uparrow q \\ \hline F & F & T \\ F & T & T \\ T & F & T \\ T & T & F \\ \hline \end{array}$

Logical NOR
The logical NOR 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 $p\ \mathsf{NOR}\ q$ (also written $p \curlywedge q$ or $p \downarrow q$) is as follows:


 * $\begin{array}{|cc||c|c|} \hline

p & q & p \downarrow q \\ \hline F & F & T \\ F & T & F \\ T & F & F \\ T & T & F \\ \hline \end{array}$