Definition:Disjunction

Definition
Disjunction is a binary connective written symbolically as $p \lor q$ whose behaviour is as follows:


 * $p \lor q$

is defined as:
 * Either $p$ is true or $q$ is true or possibly both.

This is called the disjunction of $p$ and $q$.

The statements $p$ and $q$ are known as the disjuncts.

$p \lor q$ is voiced:
 * $p$ or $q$

Boolean Interpretation
From the above, we see that the boolean interpretations for $\mathbf A \lor \mathbf B$ under the model $\mathcal M$ are:


 * $\left({\mathbf A \lor \mathbf B}\right)_{\mathcal M} = \begin{cases}

T & : \mathbf A_{\mathcal M} = T \text{ or } \mathbf B_{\mathcal M} = T \\ F & : \text {otherwise} \end{cases}$

Complement
The complement of $\lor$ is the NOR operator.

Truth Function
The disjunction connective defines the truth function $f^\lor$ as follows:

Truth Table
The truth table of $p \lor q$ and its complement is as follows:

$\begin{array}{|cc||c|c|} \hline p & q & p \lor q & p \downarrow q \\ \hline F & F & F & T \\ F & T & T & F \\ T & F & T & F \\ T & T & T & F \\ \hline \end{array}$

Notational Variants
Various symbols are encountered that denote the concept of disjunction:

Also known as
The symbol $\lor$ comes from the first letter of the Classical Latin vel.

This usage of or, that allows the case where both disjuncts are true, is called inclusive or, or the inclusive disjunction.

$p \lor q$ is also called the logical alternation of $p$ and $q$.

Also see

 * Exclusive Or