# 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:

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.

Corollaries: 1

Branches: 1
Corollaries: 2 3
Definitions: 4 5 6 7
Examples: 8
Lemmas: 9 10 11 12 13 14 15
Parts: 16
Proofs: 17 18 19 20 21 22 23 24 25 26 27 28 29
Propositions: 30
Sections: 31

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