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.