Definition:Disjunction

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


 * $$p \or q$$ 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.

"$$p \or q$$ is voiced "$$p$$ or $$q$$".

The symbol $$\or$$ is also known as vel or vee.

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


 * $$\left({\mathbf{A} \or \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 $$\or$$ is the NOR operator.

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

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

Notational Variants
Alternative symbols that mean the same thing as $$p \or q$$ are also encountered, for example:
 * $$p\ \texttt{OR}\ q$$
 * $$p + q$$

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.