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:

Truth Table of the Negation

$[[x]]_I$ $[[\neg x]]_I$
\(1\) \(0\)
\(0\) \(1\)

Notes

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


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References

Bibliography

  1. Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016