diff options
Diffstat (limited to 'ss2016/lds/01/H1-3.md')
-rw-r--r-- | ss2016/lds/01/H1-3.md | 33 |
1 files changed, 33 insertions, 0 deletions
diff --git a/ss2016/lds/01/H1-3.md b/ss2016/lds/01/H1-3.md new file mode 100644 index 0000000..b610a02 --- /dev/null +++ b/ss2016/lds/01/H1-3.md | |||
@@ -0,0 +1,33 @@ | |||
1 | a) Disjunkte Mengen bilden ein Gegenbeispiel: | ||
2 | |||
3 | \begin{align*} | ||
4 | A &= \{ a \} \\ | ||
5 | B &= \{ b \} \\ | ||
6 | C &= \{ c \} \\ | ||
7 | D &= \{ d \} \\ | ||
8 | A - (B - (C - D)) &= \{ a \} \\ | ||
9 | (A \cup C) - (B \cup D) &= \{ a, c \} | ||
10 | \end{align*} | ||
11 | |||
12 | <!-- Es sei $\Omega = A \cup B \cup C \cup D$ --> | ||
13 | |||
14 | <!-- \begin{align*} --> | ||
15 | <!-- \ & A - (B - (C - D)) = (A \cup C) - (B \cup D) \\ --> | ||
16 | <!-- \underset{\text{Extens.}}{\lequiv}& \forall x \in \Omega .\ \iota_A(x) \land \lnot (\iota_B(x) \land \lnot (\iota_C(x) \land \lnot \iota_D(x))) \lequiv (\iota_A(x) \lor \iota_C(x)) \land \lnot (\iota_B(x) \lor \iota_D(x)) \\ --> | ||
17 | <!-- \lequiv& \forall x \in \Omega .\ \iota_A(x) \land (\lnot \iota_B(x) \lor (\iota_C(x) \land \lnot \iota_D(x))) \lequiv (\iota_A(x) \lor \iota_C(x)) \land (\lnot \iota_B(x) \land \lnot \iota_D(x)) \\ --> | ||
18 | <!-- \lequiv& \forall x \in \Omega .\ (\iota_A(x) \land \lnot \iota_B(x)) \lor (\iota_A(x) \land \iota_C(x) \land \lnot \iota_D(x))) \lequiv (\iota_A(x) \land \lnot \iota_B(x) \land \lnot \iota_D(x)) \lor (\lnot \iota_B(x) \land \iota_C(x) \land \lnot \iota_D(x)) --> | ||
19 | <!-- \end{align*} --> | ||
20 | |||
21 | b) Es sei $\Omega = A \cup B$ und für eine Menge $X$ sei $\bar X = \Omega - X$. | ||
22 | $\Omega$ bildet mit $\cap, \cup$ und dem soeben definierten Komplement eine boolsche Algebra. | ||
23 | |||
24 | Die Differenz $A - B$ ist definiert als $A \cap \bar B$. | ||
25 | |||
26 | \begin{align*} | ||
27 | (A - B) \cup (B - A) &\underset{\text{\scriptsize Diff.}}{=} (A \cap \bar B) \cup (B \cap \bar A) \\ | ||
28 | &\underset{\text{\scriptsize Dist.}}{=} ((A \cap \bar B) \cup B) \cap ((A \cap \bar B) \cup \bar A) \\ | ||
29 | &\underset{\text{\scriptsize Dist.}}{=} ((A \cup B) \cap (\bar B \cup B)) \cap ((A \cup \bar A) \cap (\bar B \cup \bar A)) \\ | ||
30 | &\underset{\text{\scriptsize Tnd.\footnotemark}}{=} (A \cup B) \cap (\bar A \cup \bar B) \\ | ||
31 | &\underset{\text{\scriptsize Diff.}}{=} (A \cup B) - (A \cap B) | ||
32 | \end{align*} | ||
33 | \footnotetext{Tertium non datur} | ||