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}$$