**Definition**: Disjunction

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:

$[[x]]_I$ | $[[y]]_I$ | $[[x\vee y]]_I$ |
---|---|---|

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

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

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

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

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.

| | | | | Contributors: *bookofproofs*

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

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

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

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