# Properties of Disjunction

Learn about the basic properties of disjunction.

## We'll cover the following

## 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.

### Associativity

Disjunction is associative. This means:

$\left(p \lor q\right) \lor r \equiv p \lor \left(q \lor r\right).$

A formal proof of associativity of disjunction can be given by making a truth table. However, it is easier and more intuitive to argue that both, $( p \lor q) \lor r$ and $p \lor (q \lor r)$, are true if and only if at least one of the three boolean variables is true. Formally, the following truth table shows that disjunction is associative. This is established because the last two columns are identical.

$p$ | $q$ | $r$ | $p\lor q$ | $q\lor r$ | $\left(p \lor q\right) \lor r$ | $p \lor \left(q \lor r\right)$ |
---|---|---|---|---|---|---|

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

The disjunction operator is also commutative. This means taking disjunction of $p$ with $q$ is the same as taking disjunction of $q$ with $p$; that is,

$p \lor q \equiv q \lor p.$

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.

## Quiz

Test your understanding of the properties of disjunction.

**(Select all that apply.)** What are the properties of disjunction?

Disjunction is commutative.

Disjunction is not commutative.

Disjunction is associative.

Disjunction is not associative.