Definition:Truth Table

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 and where $$\mathbb{B}$$ is the boolean domain $$\{ 0, 1 \}.\!$$

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 $$\sim p$$ or $$\neg p$$) is as follows:

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:

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 $$p\wedge q$$, $$p$$&$$q$$, or $$p\cdot q$$) is as follows:

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 $$p \vee q$$) is as follows:

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 $$p = q$$, $$p \Leftrightarrow q$$, or $$p \equiv q$$) is as follows:

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 $$p + q$$, $$p \oplus q$$, or $$p \neq q$$) is as follows:

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 $$p\rightarrow q$$) and the logical implication p implies q (symbolized as $$p\implies q$$) is as follows:

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 $$p\uparrow q$$) is as follows:

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 $$p \bot q$$ or $$p\downarrow q$$) is as follows: