Definition:Disjunction/General Definition

Definition

Let $p_1, p_2, \ldots, p_n$ be statements.

The disjunction of $p_1, p_2, \ldots, p_n$ is defined as:

$\ds \bigvee_{i \mathop = 1}^n \ p_i = \begin{cases}

p_1 & : n = 1 \\ & \\ \ds \paren {\bigvee_{i \mathop = 1}^{n - 1} \ p_i} \lor p_n & : n > 1 \end{cases}$

That is:

$\ds \bigvee_{i \mathop = 1}^n \ p_i = p_1 \lor p_2 \lor \cdots \lor p_{n - 1} \lor p_n$

In terms of the set $P = \set {p_1, \ldots, p_n}$ this can also be rendered:

$\ds \bigvee P$

and is referred to as the disjunction of $P$.