Here is an Exercise from the following book https://www.amazon.fr/Mathématiques-Méthodes-Exercices-MPSI-programme/dp/210072780X/ref=pd_sim_14_1?_encoding=UTF8&psc=1&refRID=6KD4X5JTZGWMTDBXP238 that i can't understand his solution specialy how the authour found These bijective maps to be able to calculate the number f parts X in differents cases.
Exercise :
Number of parts or pairs of parts verifying conditions.
Let $E$ be a finite set, $n = |E|, A ⊂ E, B ⊂ E, p = |A|, q = |B|, r = |A\cap B|$.
- Determine the number of parts X of E such that:
1 $X \cup A = E$
2 .$A\cap B \subset X \subset A\cup B$
Solutions provided by the book:
The map $$Z\longmapsto \bar{A} \cup Z $$ is a bijection of the set of All the parts $Z$ of $A$ on the set of the parts $X$ of $E$ Such that $X \cup A = E$. The number sought is therefore the number of parts of $A$, that is $2^{p}$.
The map $$Z\longmapsto (A\cap B ) \cup Z $$ is a bijection of the set of parts of $(A\cup B )\setminus (A\cap B)$ onto the set of parts X of E such that $A\cap B \subset X \subset A\cup B$, so the number sought is the number of parts of $(A\cup B )\setminus (A\cap B)$ . Since A ∩ B ⊂ A ∪ B, we have:
$$2^{card\left(A\cap B \subset X \subset A\cup B\right)}=2^{p+q-2r}$$
My questions:
- Could someone explain to me how the author of the book found these bijective maps to calculate the number of parts in differents cases.
- if isn't obvious should i memorize them
My thoughts about how the author found those bijection maps above
for case : $X \cup A = E$
$\forall X\subset E$; \begin{aligned} X\cup A=E &\iff \overline{X\cup A}=\emptyset \\ &\iff \overline{X} \subset A \end{aligned} thus we have the the following bijective function:
\begin{align*} f: \{ X\in\mathcal{P}\left(A\right), X\cup A=E \} &\rightarrow \{Y\in \mathcal{P}\left(E \right), Y\subset A \}\\ X&\mapsto \overline{X} \end{align*} i found bijection map but not the same as worte the author of the book
$P(X)$ means Power set of a set $X$