Aufgabe 257
Aufgaben zur Mengenlehre
1. Teilaufgabe - Bearbeitungszeit 05:40
Schreibe die Elemente an, die in den jeweiligen Mengen enthalten sind.
\(\eqalign{ & {M_1} = \left\{ {x \in {N^ + }|x < 7} \right\} \cr & {M_2} = \left\{ {x \in N|7 < x \leqslant 9} \right\} \cr & {M_3} = \left\{ {x \in Z| - 3 < x < 2} \right\} \cr & {M_4} = \left\{ {x \in Z| - 3 < x} \right\} \cr & {M_5} = \left\{ {x \in Z| - 3 < x < - 2} \right\} \cr & {M_6} = \left\{ {x \in N|8 \leqslant x \leqslant 9} \right\} \cr} \)
2. Teilaufgabe - Bearbeitungszeit 05:40
Setze das gegebene Element in Beziehung zur Menge unter Verwendung von \( \in ,\,\, \notin ,\,\, \subset ,\,\, \subseteq \)
\(\eqalign{ & 2\_?\_{M_1} \cr & 7\_?\_{M_1} \cr & 2\_?\_{M_5} \cr & {M_3}\_?\_{M_4} \cr & {M_2}\_?\_{M_6} \cr} \)
3. Teilaufgabe - Bearbeitungszeit 05:40
Schreibe die Durchschnittsmenge an
\(\eqalign{ & {M_1} \cap {M_2} \cr & {M_1} \cap {M_3} \cr & {M_1} \cap {M_4} \cr & {M_3} \cap {M_4} \cr & {M_4} \cap {M_6} \cr} \)
4. Teilaufgabe - Bearbeitungszeit 05:40
Schreibe die Vereinigungsmenge an
\(\eqalign{ & {M_1} \cup {M_2} \cr & {M_2} \cup {M_3} \cr & {M_5} \cup {M_6} \cr & {M_4} \cup {M_6} \cr & {M_1} \cup {M_4} \cr} \)
5. Teilaufgabe - Bearbeitungszeit 05:40
Schreibe die Differenzmenge an
\(\eqalign{ & {M_1}\backslash {M_2} \cr & {M_1}\backslash {M_3} \cr & {M_3}\backslash {M_1} \cr & {M_2}\backslash {M_5} \cr & {M_4}\backslash {M_3} \cr} \)
Lösungsweg
1. Teilaufgabe
\(\eqalign{ & {M_1} = \left\{ {x \in {N^ + }|x < 7} \right\} \Rightarrow {M_1} = \left\{ {1,2,3,4,5,6} \right\} \cr & {M_2} = \left\{ {x \in N|7 < x \leqslant 9} \right\} \Rightarrow {M_2} = \left\{ {8,9} \right\} \cr & {M_3} = \left\{ {x \in Z| - 3 < x < 2} \right\} \Rightarrow {M_3} = \left\{ { - 2, - 1,0,1} \right\} \cr & {M_4} = \left\{ {x \in Z| - 3 < x} \right\} \Rightarrow {M_4} = \left\{ { - 2, - 1,0,1,2,...\infty } \right\} \cr & {M_5} = \left\{ {x \in Z| - 3 < x < - 2} \right\} \Rightarrow {M_5} = \left\{ {} \right\} \cr & {M_6} = \left\{ {x \in N|8 \leqslant x \leqslant 9} \right\} \Rightarrow {M_6} = \left\{ {8,9} \right\} \cr} \)
2. Teilaufgabe
\(\eqalign{ & 2\_?\_{M_1} \Rightarrow 2 \in {M_1} \cr & 7\_?\_{M_1} \Rightarrow 7 \notin {M_1} \cr & 2\_?\_{M_5} \Rightarrow 2 \notin {M_5} \cr & {M_3}\_?\_{M_4} \Rightarrow {M_3} \subseteq {M_4} \cr & {M_2}\_?\_{M_6} \Rightarrow {M_2} \subset {M_6} \cr} \)
3. Teilaufgabe
\(\eqalign{ & {M_1} \cap {M_2} \Rightarrow \left\{ {} \right\} \cr & {M_1} \cap {M_3} \Rightarrow \left\{ 1 \right\} \cr & {M_1} \cap {M_4} \Rightarrow \left\{ {1,2,3,4,5,6} \right\} \cr & {M_3} \cap {M_4} \Rightarrow \left\{ { - 2, - 1,0,1} \right\} \cr & {M_4} \cap {M_6} \Rightarrow {M_4} \cr} \)
4. Teilaufgabe
\(\eqalign{ & {M_1} \cup {M_2} \Rightarrow \left\{ {1,2,3,4,5,6,8,9} \right\} \cr & {M_2} \cup {M_3} \Rightarrow \left\{ { - 2, - 1,0,1,8,9} \right\} \cr & {M_5} \cup {M_6} \Rightarrow {M_6} \cr & {M_4} \cup {M_6} \Rightarrow {M_4} \cr & {M_1} \cup {M_4} \Rightarrow {M_4} \cr} \)
5. Teilaufgabe
\(\eqalign{ & {M_1}\backslash {M_2} \Rightarrow \left\{ {} \right\} \cr & {M_1}\backslash {M_3} \Rightarrow \left\{ {2,3,4,5,6} \right\} \cr & {M_3}\backslash {M_1} \Rightarrow \left\{ { - 2, - 1,0} \right\} \cr & {M_2}\backslash {M_5} \Rightarrow \left\{ {8,9} \right\} \cr & {M_4}\backslash {M_3} \Rightarrow \left\{ {2,3,...\infty } \right\} \cr} \)
Ergebnis
Die richtige Lösung lautet
1. Teilaufgabe
\(\eqalign{ & {M_1} = \left\{ {1,2,3,4,5,6} \right\} \cr & {M_2} = \left\{ {8,9} \right\} \cr & {M_3} = \left\{ { - 2, - 1,0,1} \right\} \cr & {M_4} = \left\{ { - 2, - 1,0,1,2,...\infty } \right\} \cr & {M_5} = \left\{ {} \right\} \cr & {M_6} = \left\{ {8,9} \right\} \cr} \)
2. Teilaufgabe
\(\eqalign{ & 2 \in {M_1} \cr & 7 \notin {M_1} \cr & 2 \notin {M_5} \cr & {M_3} \subseteq {M_4} \cr & {M_2} \subset {M_6} \cr} \)
3. Teilaufgabe
\(\eqalign{ & \left\{ {} \right\} \cr & \left\{ 1 \right\} \cr & \left\{ {1,2,3,4,5,6} \right\} \cr & \left\{ { - 2, - 1,0,1} \right\} \cr & {M_4} \cr} \)
4. Teilaufgabe
\(\eqalign{ & \left\{ {1,2,3,4,5,6,8,9} \right\} \cr & \left\{ { - 2, - 1,0,1,8,9} \right\} \cr & {M_6} \cr & {M_4} \cr & {M_4} \cr} \)
5. Teilaufgabe
\(\eqalign{ & \left\{ {} \right\} \cr & \left\{ {2,3,4,5,6} \right\} \cr & \left\{ { - 2, - 1,0} \right\} \cr & \left\{ {8,9} \right\} \cr & \left\{ {2,3,...\infty } \right\} \cr} \)