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 (or logical alternation) 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:

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

Compare exclusive or.