Accueil > Publications > Replace Numbers with Sets in calculations (1).

Replace Numbers with Sets in calculations (1).

mardi 22 janvier 2019, par ScientificWare

Why it’s so important to calculate with sets ?

Some calculations are not defined with numbers but have a result with sets :
- $\frac{\lbrace 3 \rbrace}{\lbrace 0 \rbrace}=\emptyset$
- $\frac{\lbrace 3 \rbrace}{\lbrace 2 \rbrace}=\emptyset$
- and $\frac{\lbrace 0 \rbrace}{\lbrace 0 \rbrace} \in \mathcal{P}(\mathbb{N}),\mathcal{P}(\mathbb{Z}),\mathcal{P}(\mathbb{D}), \mathcal{P}(\mathbb{Q}), \mathcal{P}(\mathbb{R}), \mathcal{P}(\mathbb{C}), ...$ depending on which set, the operation / is defined.
Computer Operating Systems return an error for such calculations. Thus we could think that consequences are similar. But interpretations are very different.
- No result in the first case.
- Many results or an infinity of results in the second case, a really tricky case to interpret, depending of the set, calculus accuracy, ... :
  - calculus accuracy,
  - set nature,
  - ...
The use of Set Calcul would force to evaluate more interpretations.
Read your courses about Lebesgue’s Integral, to see why it’s so powerful ?