Dmitry
1.2 KiB
1.2 KiB
status, type, tags, created, updated, title
| status | type | tags | created | updated | title | ||
|---|---|---|---|---|---|---|---|
| seed | concept |
|
2025-12-17 | 2026-05-07 | Доказательство законов Де Моргана |
Доказательство законов Де Моргана
\overline{A \cup B} = \overline{A} \cap \overline{B}
1 - \mu_{A \cup B}(u)=1 - \max (\mu_{A}(u);\ \mu_{B}(u))
\overline{A} \cap \overline{B} = \min(1-\mu_{A}(u);\ 1-\mu_{B}(u))
Предположим, что \mu_{B}(u) > \mu_{A}(u), тогда 1-\mu_{B}(u)=1-\mu_{B}(u)
\overline{A \cap B} = \overline{A} \cup \overline{B}
1-\mu_{A \cap B}=1 -min(\mu_{A}(u);\ \mu_{B}(u))
\overline{A} \cup \overline{B} = max(1 - \mu_{A}(u);\ 1-\mu_{B}(u))
Пусть \mu_{B}(u)>\mu_{A}(u), тогда 1-\mu_{A}(u)=1-\mu_{A}(u)
\overline{A * B} = \overline{A} + \overline{B}
(\mu_{\overline{A}}(u) + \mu_{\overline{B}}(u) - \mu_{\overline{A}}(u)\mu_{\overline{B}}(u)) =
= 1-\mu_{A}(u) + 1 - \mu_{B}(u) - 1 + \mu_{A}(u) + \mu_{B}(u)-\mu_{A}(u)\mu_{B}(u)=
= 1 - \mu_{A}(u) \mu_{B}(u)
\overline{A + B} = \overline{A} * \overline{B}
(1-\mu_{A}(u))*(1-\mu_{B}(u)) = 1-\mu_{A}(u)-\mu_{B}(u)+\mu_{A}(u)\mu_{B}(u) =
1 - (\mu_{A}(u)+\mu_{B}(u)-\mu_{A}(u)\mu_{B}(u))