After we have explained what Boolean functions and what truth tables are, it is time to use these concepts to introduce important examples of connectives.

**Definition**: Negation

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

\[\neg :=\begin{cases}L & \mapsto L \\

x &\mapsto \neg x \\

\end{cases}\]

defined by the following truth table:

$[[x]]_I$ | $[[\neg x]]_I$ |
---|---|

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

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

### Notes

- The negation of any proposition changes from one truth value to the other.
- By the axiom of bivalence of truth, the negation of any negated proposition $\neg x$ must be the proposition itself: $\neg(\neg x)=x.$
- Later in the text, it will turn out that negation is one of the most important connectives: We can construct successfully logical systems even if we dispense some connectives, but we cannot do without the negation.

| | | | | Contributors: *bookofproofs* | References: [7838]

(none)

[7838] **Kohar, Richard**: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016

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