Let \(L\) be a set of propositions with the interpretation $I$ and the corresponding valuation function $[[]]_I$. A **disjunction** “\(\vee\)” is a Boolean function

$$\vee :=\begin{cases}

L \times L & \mapsto L \\

(x,y) &\mapsto x \vee y.

\end{cases}$$

defined by the following truth table:

Truth Table of the Conjunction
$[[x]]_I$ |
$[[y]]_I$ |
$[[x\vee y]]_I$ |

\(1\) |
\(1\) |
\(1\) |

\(0\) |
\(1\) |
\(1\) |

\(1\) |
\(0\) |
\(1\) |

\(0\) |
\(0\) |
\(0\) |

We read the disjunction $x\vee y$

“$x$ *or* $y$”.

### Notes

- The disjunction of two propositions is only false if the two propositions are both false. Otherwise, it is true.
- The word
*or* in the natural English language does not correspond to the logical *or* disjunction. Consider the following example:

$x=$“The month is January”, $y=$“The month is February”

The logical disjunction $x\vee y$ is true if one of $x,y$ (or both) are true, but in English, only one can be true, but never both.

| | | | | created: 2014-06-22 20:32:11 | modified: 2018-05-23 23:08:18 | by: *bookofproofs*

## 1.**Corollary**: Commutativity of Disjunction

## 2.**Proposition**: Associativity of Disjunction

## 3.**Definition**: Exclusive Disjunction