Properties of Disjunction

Properties

Just like arithmetic operations satisfy specific properties, logical operations like disjunction also satisfy certain properties. Here, we will discuss the most fundamental properties of disjunction.

$p$	$q$	$r$	$p\lor q$	$q\lor r$	$\left(p \lor q\right) \lor r$	$p \lor \left(q \lor r\right)$
T	T	T	T	T	T	T
T	T	F	T	T	T	T
T	F	T	T	T	T	T
T	F	F	T	F	T	T
F	T	T	T	T	T	T
F	T	F	T	T	T	T
F	F	T	F	T	T	T
F	F	F	F	F	F	F

Commutativity

Given $n$ propositions, $p_{1},p_{2},...,p_{n},$ we can take their disjunction and construct a new proposition $p$ as follows:

$$p \equiv p_{1}\lor p_{2}\lor \cdots \lor p_{n} .$$

A convenient way to represent a disjunction of many variables is the iterative notation. In this notation, the above disjunction is written as:

$$p \equiv \bigvee_{i=1}^n p_i.$$

With commutativity and associativity in hand, we can arrange these $n$ propositions on the right-hand side of the equation above in any of $n!$ possible ways. Furthermore, we can parenthesize the conjunction in many possible ways and the resulting proposition will remain equivalent to $p$. The proposition $p$ will only be true if at least one of the $n$ propositions, $p_{1}$, $p_{2}$, …, $p_{n}$, is true . If all the $n$ propositions are false then $p$ will be false.

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