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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.

| | | | | Contributors: bookofproofs

1.Corollary: Commutativity of Disjunction

2.Proposition: Associativity of Disjunction

3.Definition: Exclusive Disjunction


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