Difference between revisions of "Theory of Stochastic Signals/Set Theory Basics"

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{{Header
 
{{Header
|Untermenü=Wahrscheinlichkeitsrechnung
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|Untermenü=Probability Calculation
|Vorherige Seite=Einige grundlegende Definitionen
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|Vorherige Seite=Some Basic Definitions
|Nächste Seite=Statistische Abhängigkeit und Unabhängigkeit
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|Nächste Seite=Statistical Dependence and Independence
 
}}
 
}}
==Venndiagramm, Grundmenge und leere Menge==
+
==Venn diagram, universal and empty set==
 
<br>
 
<br>
[[File:EN_Sto_T_1_2_S1.png | right|frame|Mengendarstellung im Venndiagramm]]
+
In later chapters,&nbsp; we will sometimes refer to&nbsp; [https://en.wikipedia.org/wiki/Set_theory &raquo;set theory&laquo;]&nbsp;.&nbsp; Therefore,&nbsp; the most important basics and definitions of this discipline will be briefly summarized here.&nbsp; The topic is also covered in the&nbsp; $($German language$)$&nbsp; learning video&nbsp; [[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)|&raquo;Mengentheoretische Begriffe und Gesetzmäßigkeiten&laquo;]] &nbsp; &rArr; &nbsp; &raquo;Set Theory &ndash; Terms and Regularities&laquo;.
In späteren Kapitel wird manchmal auf die&nbsp; [https://de.wikipedia.org/wiki/Mengenlehre Mengenlehre]&nbsp; Bezug genommen.&nbsp; Deshalb sollen hier die wichtigsten Grundlagen und Definitionen dieser Disziplin kurz zusammengefasst werden.&nbsp; Die Thematik wird auch im Lernvideo&nbsp; [[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)|Mengentheoretische Begriffe und Gesetzmäßigkeiten]]&nbsp; am Beispiel europäischer Staaten behandelt.
+
[[File:EN_Sto_T_1_2_S1.png | right|frame|Set representation in the Venn diagram]]
  
Ein wichtiges Hilfsmittel der Mengenlehre ist das&nbsp; '''Venndiagramm'''&nbsp; gemäß der Grafik:
+
An important tool of set theory is the&nbsp; &raquo;'''Venn diagram'''&laquo;&nbsp; according to the graph:
*Angewandt auf die Wahrscheinlichkeitsrechnung sind hier die Ereignisse&nbsp; $A_i$&nbsp; als Flächenbereiche dargestellt.&nbsp; Zur einfacheren Beschreibung bezeichnen wir hier die Ereignisse im Gegensatz zum letzten Kapitel nicht mit&nbsp; $A_1$,&nbsp; $A_2$&nbsp; und&nbsp;  $A_3$, sondern mit&nbsp; $A$,&nbsp; $B$&nbsp; und&nbsp; $C$. Die Gesamtfläche entspricht der Grundmenge&nbsp; $G$.
+
*Applied to probability theory,&nbsp; the events&nbsp; $A_i$&nbsp; are represented here as areas.&nbsp; For a simpler description we do not denote the events here with&nbsp; $A_1$,&nbsp; $A_2$&nbsp; and&nbsp;  $A_3$,&nbsp; but with&nbsp; $A$,&nbsp; $B$&nbsp; and&nbsp; $C$ in contrast to the last chapter.&nbsp;  
*Die Grundmenge&nbsp; $G$&nbsp; beinhaltet alle möglichen Ergebnisse und steht für das&nbsp; '''Sichere Ereignis''', das definitionsgemäß mit der Wahrscheinlichkeit „Eins” eintritt:  &nbsp; ${\rm Pr}(G) = 1$. Zum Beispiel ist beim Zufallsexperiment „Werfen eines Würfels”  die Wahrscheinlichkeit für das Ereignis „die Augenzahl ist kleiner oder gleich 6” identisch Eins.
 
*Dagegen beinhaltet die&nbsp; '''Leere Menge'''&nbsp; $ϕ$&nbsp; kein einziges Element.&nbsp; Bezogen auf Ereignisse gibt die Leere Menge das&nbsp; '''Unmögliche Ereignis'''&nbsp; mit der Wahrscheinlichkeit&nbsp; ${\rm Pr}(ϕ) = 0$&nbsp; an.&nbsp; Beispielsweise ist beim Experiment „Werfen eines Würfels”'  die Wahrscheinlichkeit für das Ereignis „die Augenzahl ist größer als 6” identisch Null.
 
  
 +
*The total area corresponds to the&nbsp; &raquo;universal set&laquo;&nbsp; $($or short:&nbsp; &raquo;universe&laquo;$)$&nbsp; $G$.&nbsp; The universe&nbsp; $G$&nbsp; contains all possible outcomes and stands for the&nbsp; &raquo;'''certain event'''&laquo;,&nbsp; which by definition occurs with probability &raquo;one&laquo;:  &nbsp; ${\rm Pr}(G) = 1$.&nbsp;  For example,&nbsp; in the random experiment&nbsp; &raquo;Throwing a die&laquo;,&nbsp; the probability for the event&nbsp; &raquo;The number of eyes is less than or equal to 6&laquo;&nbsp; is identical to one.
  
Anzumerken ist, dass nicht jedes Ereignis&nbsp; $A$&nbsp; mit&nbsp; ${\rm Pr}(A) = 0$&nbsp; wirklich nie eintreten kann:
+
*In contrast,&nbsp; the&nbsp; &raquo;'''empty set'''&laquo;&nbsp; $ϕ$&nbsp; does not contain a single element.&nbsp; In terms of events,&nbsp; the empty set specifies the&nbsp; &raquo;'''impossible event'''&laquo;&nbsp; with probability&nbsp; ${\rm Pr}(ϕ) = 0$&nbsp; an.&nbsp; For example,&nbsp; in the experiment&nbsp; &raquo;Throwing a die&laquo;,&nbsp; the probability for the event&nbsp; &raquo;The number of eyes is greater than 6&laquo;&nbsp; is identically zero.  
*So ist die Wahrscheinlichkeit des Ereignisses „der Rauschwert&nbsp; $n$&nbsp; ist identisch Null” zwar verschwindend klein und es gilt&nbsp; ${\rm Pr}(n \equiv 0) = 0$, wenn&nbsp; $n$&nbsp; durch eine kontinuierliche (Gaußsche) Zufallsgröße beschrieben wird.
 
*Trotzdem ist es natürlich möglich (wenn auch extrem unwahrscheinlich), dass irgendwann auch der exakte Rauschwert&nbsp; $n = 0$&nbsp; auftritt.
 
  
==Vereinigungsmenge==
+
 
 +
It should be noted that not every event&nbsp; $A$&nbsp; with&nbsp; ${\rm Pr}(A) = 0$&nbsp; can really never occur:
 +
*Thus,&nbsp; the probability of the event&nbsp; &raquo;the noise value&nbsp; $n$&nbsp; is identical to zero&raquo;&nbsp; is vanishingly small and it applies&nbsp; ${\rm Pr}(n \equiv 0) = 0$,&nbsp;  if&nbsp; $n$&nbsp; is described by a continuous&ndash;valued&nbsp; $($Gaussian$)$&nbsp; random variable.
 +
 
 +
*Nevertheless,&nbsp; it is of course possible&nbsp; $($although extremely unlikely$)$&nbsp; that at some points the exact noise value&nbsp; $n = 0$&nbsp; will also occur.
 +
 
 +
==Union set==
 
<br>
 
<br>
Anhand des Venndiagramms werden nun einige mengentheoretische Verknüpfungen erläutert.
+
Some set-theoretical relations are explained now on the basis of the Venn diagram.
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definition:}$&nbsp; Die&nbsp; '''Vereinigungsmenge'''&nbsp; $C$&nbsp; zweier Mengen&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; beinhaltet alle die Elemente, die entweder in der Menge&nbsp; $A$&nbsp; oder der Menge&nbsp; $B$&nbsp; oder in beiden enthalten sind&nbsp; (englisch: ''Set Union'' ).&nbsp; Formelmäßig wird dieser Zusammenhang wie folgt ausgedrückt:
+
$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''union set'''&laquo;&nbsp; $C$&nbsp; of two sets&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; contains all the elements that are contained either in set&nbsp; $A$&nbsp; or in set&nbsp; $B$&nbsp; or in both.&nbsp;
$$\ C = A \cup B \hspace{0.2cm}(= A + B).$$
+
[[File: EN_Sto_T_1_2_S2.png  |right|frame| Union set in the Venn diagram]]
 +
*This relationship is expressed as the following formula:
 +
:$$\ C = A \cup B.$$
  
In der Literatur ist auch die Bezeichnung&nbsp; ''Summenmenge''&nbsp; gebräuchlich und es wird manchmal das Pluszeichen benutzt.&nbsp; In unserem Tutorial verwenden wir jedoch ausschließlich das&nbsp; $$-Zeichen.}}
+
*Using the diagram, it is easy to see the following laws of set theory:
 +
:$$A \cup \it \phi = A \rm \hspace{3.6cm}(union \hspace{0.15cm}with \hspace{0.15cm}the \hspace{0.15cm}empty \hspace{0.15cm}set),$$
 +
:$$A\cup G = G \rm \hspace{3.6cm}(union \hspace{0.15cm}with \hspace{0.15cm}the \hspace{0.15cm}universe),$$
 +
:$$A\cup A = A  \hspace{3.6cm}(\rm tautology),$$
 +
:$$A\cup B = B\cup A \hspace{2.75cm}(\rm commutative \hspace{0.15cm}property),$$
 +
:$$(A\cup B)\cup C = A\cup (B\cup C) \hspace{0.45cm}(\rm associative \hspace{0.15cm}property).$$
  
 +
*If nothing else is known about the event sets&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; then only a lower bound and an upper bound can be given for the probability of the union set:
 +
:$${\rm Max}\big({\rm Pr} (A), \ {\rm Pr} (B)\big) \le {\rm Pr} (A \cup B) \le  {\rm Pr} (A)+{\rm Pr} (B).$$
  
[[File: EN_Sto_T_1_2_S2.png  |right|frame| Vereinigungsmenge im Venndiagramm]]
+
*The probability of the union set is equal to the lower bound if&nbsp; $A$&nbsp; is a&nbsp; [[Theory_of_Stochastic_Signals/Set_Theory_Basics#Proper_subset_.E2.80.93_Improper_subset|$\text{subset}$]]&nbsp; of&nbsp; $B$&nbsp; or vice versa.  
Anhand der Grafik sind die folgenden Gesetzmäßigkeiten der Mengenlehre leicht einzusehen:
 
:$$A \cup \it \phi = A \rm \hspace{3.6cm}(Vereinigung \hspace{0.15cm}mit \hspace{0.15cm}der \hspace{0.15cm}leeren \hspace{0.15cm}Menge),$$
 
:$$A\cup G = G \rm \hspace{3.6cm}(Vereinigung \hspace{0.15cm}mit \hspace{0.15cm}der \hspace{0.15cm}Grundmenge),$$
 
:$$A\cup A = A  \hspace{3.6cm}(\rm Tautologiegesetz),$$
 
:$$A\cup B = B\cup A \hspace{2.75cm}(\rm Kommutativgesetz),$$
 
:$$(A\cup B)\cup C = A\cup (B\cup C) \hspace{0.45cm}(\rm Assoziativgesetz).$$
 
  
Ist über die Ereignismengen&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; nichts weiter bekannt, so können für die Wahrscheinlichkeit der Vereinigungsmenge nur eine untere und eine obere Schranke angegeben werden:
+
*The upper bound holds for&nbsp; [[Theory_of_Stochastic_Signals/Set_Theory_Basics#Disjoint_sets|&raquo;disjoint sets&laquo;]].}}
:$${\rm Max}\big({\rm Pr} (A), \ {\rm Pr} (B)\big) \le {\rm Pr} (A \cup B) \le  {\rm Pr} (A)+{\rm Pr} (B).$$
 
  
*Die Wahrscheinlichkeit der Vereinigungsmenge ist gleich der unteren Schranke, wenn&nbsp; $A$&nbsp; eine&nbsp; [[Theory_of_Stochastic_Signals/Mengentheoretische_Grundlagen#Echte_Teilmenge_.E2.80.93_unechte_Teilmenge|Teilmenge]]&nbsp; von&nbsp; $B$&nbsp; ist oder umgekehrt.
 
*Die obere Schranke gilt für&nbsp; [[Theory_of_Stochastic_Signals/Mengentheoretische_Grundlagen#Disjunkte_Mengen|disjunkte Mengen]].
 
  
 +
{{GraueBox|TEXT= 
 +
$\text{Example 1:}$&nbsp; We consider again the experiment&nbsp; &raquo;throwing a die&laquo;.&nbsp; The possible outcomes&nbsp; $($number of points$)$&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
  
{{GraueBox|TEXT= 
+
Consider the two events
$\text{Beispiel 1:}$&nbsp; Betrachtet man die beiden Ereignisse
+
* $A :=$&nbsp; &raquo;The outcome is greater than or equal to&nbsp; $5$ &laquo;&nbsp; $ = \{5, 6\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A)= 2/6= 1/3$,
* $A :=$ „die Augenzahl ist größer oder gleich $5$$ = \{5, 6\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A)= 2/6= 1/3$,  
+
* $B :=$ „die Augenzahl ist geradzahlig” $= \{2, 4, 6\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (B)= 3/6= 1/2$,  
+
* $B :=$&nbsp; &raquo;The outcome is even &laquo;&nbsp; $= \{2, 4, 6\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (B)= 3/6= 1/2$,  
  
  
so beinhaltet die Vereinigungsmenge vier Elemente: &nbsp;  $(A \cup B) = \{2, 4, 5, 6 \}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A \cup B) = 4/6 = 2/3$.  
+
then the union set contains four elements: &nbsp;  $(A \cup B) = \{2, 4, 5, 6 \}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A \cup B) = 4/6 = 2/3$.  
*Die untere Schranke ergibt sich zu&nbsp; ${\rm Pr} (A \cup B) \ge {\rm Max}\big({\rm Pr} (A),\ {\rm Pr} (B)\big ) = 3/6.$
+
*For the lower bound: &nbsp; ${\rm Pr} (A \cup B) \ge {\rm Max}\big({\rm Pr} (A),\ {\rm Pr} (B)\big ) = 3/6.$
*Für die obere Schranke gilt&nbsp; $ {\rm Pr} (A \cup B) \le  {\rm Pr} (A)+{\rm Pr} (B) = 5/6.$}}
+
*For the upper bound: &nbsp; $ {\rm Pr} (A \cup B) \le  {\rm Pr} (A)+{\rm Pr} (B) = 5/6.$}}
  
==Schnittmenge==
+
==Intersection set==
 
<br>
 
<br>
Eine weitere wichtige mengentheoretische Verknüpfung stellt die Schnittmenge dar.
+
Another important set-theoretic relation is the intersection.
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definition:}$&nbsp; Die&nbsp; '''Schnittmenge'''&nbsp; $C$&nbsp; zweier Mengen&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; beinhaltet alle diejenigen Elemente, die sowohl in der Menge&nbsp; $A$&nbsp; als auch in der Menge&nbsp; $B$&nbsp; enthalten sind&nbsp; (englisch: ''Intersecting Set''  ). Formelmäßig wird dieser Zusammenhang wie folgt ausgedrückt:
+
$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''intersection set'''&laquo;&nbsp; $C$&nbsp; of two sets&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; contains all those elements which are contained in both the set&nbsp; $A$&nbsp; and the set&nbsp; $B$.
:$$C = A \cap B \hspace{0.2cm}(= A \cdot B).$$
+
[[File:EN_Sto_T_1_2_S3.png |right|frame| Intersection set in the Venn diagram]]
  
In der Literatur ist hierfür auch die Bezeichnung ''Produktmenge''&nbsp; gebräuchlich und man verwendet teilweise das Multiplikationssymbol. }}
+
*This relationship is expressed as the following formula:
 +
:$$C = A \cap B.$$
  
 +
*In the diagram,&nbsp; the intersection is shown in purple.&nbsp; Analog to the union set,&nbsp; the following regularities apply here:
 +
:$$A \cap \it \phi = \it \phi \rm \hspace{3.75cm}(intersection \hspace{0.15cm}with \hspace{0.15cm}the \hspace{0.15cm}empty \hspace{0.15cm}set),$$
 +
:$$A \cap G = A \rm \hspace{3.6cm}(intersection \hspace{0.15cm}with \hspace{0.15cm}the \hspace{0.15cm}universe),$$
 +
:$$A\cap A = A  \rm \hspace{3.6cm}(tautology),$$
 +
:$$A\cap B = B\cap A \rm \hspace{2.75cm}(commutative \hspace{0.15cm}property),$$
 +
:$$(A\cap B)\cap C = A\cap (B\cap C) \rm \hspace{0.45cm}(associative \hspace{0.15cm}property).$$
  
[[File:EN_Sto_T_1_2_S3.png |right|frame| Schnittmenge im Venndiagramm]]
+
*If nothing else is known about&nbsp; $A$&nbsp; and&nbsp; $B$,&nbsp; then no statement can be made for the probability of the intersection.
In der Grafik ist die Schnittmenge violett dargestellt.&nbsp; Analog zur Vereinigungsmenge sind hier folgende Gesetzmäßigkeiten zu nennen:
+
 
:$$A \cap \it \phi = \it \phi \rm \hspace{3.75cm}(Schnitt \hspace{0.15cm}mit \hspace{0.15cm}der \hspace{0.15cm}leeren \hspace{0.15cm}Menge),$$
+
*However,&nbsp;  if&nbsp; ${\rm Pr} (A) \le 1/2$&nbsp; and at the same time&nbsp; ${\rm Pr} (B) \le 1/2$ hold,&nbsp; then a lower and an upper bound can be given:
:$$A \cap G = A \rm \hspace{3.6cm}(Schnitt \hspace{0.15cm}mit \hspace{0.15cm}der \hspace{0.15cm}Grundmenge),$$
 
:$$A\cap A = A  \rm \hspace{3.6cm}(Tautologiegesetz),$$
 
:$$A\cap B = B\cap A \rm \hspace{2.75cm}(Kommutativgesetz),$$
 
:$$(A\cap B)\cap C = A\cap (B\cap C) \rm \hspace{0.45cm}(Assoziativgesetz).$$
 
<br clear=all>
 
*Ist über&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; nichts weiter bekannt, so kann für die Wahrscheinlichkeit der Schnittmenge keine Aussage getroffen werden.  
 
*Gilt jedoch&nbsp; ${\rm Pr} (A) \le 1/2$&nbsp; und gleichzeitig&nbsp; ${\rm Pr} (B) \le 1/2$, so kann eine untere und eine obere Schranke angegeben werden:
 
 
:$$0 \le {\rm Pr} (A ∩ B) \le {\rm Min}\ \big({\rm Pr} (A),\  {\rm Pr} (B)\big ).$$
 
:$$0 \le {\rm Pr} (A ∩ B) \le {\rm Min}\ \big({\rm Pr} (A),\  {\rm Pr} (B)\big ).$$
  
*${\rm Pr}(A ∩ B)$&nbsp; wird manchmal auch "Verbundwahrscheinlichkeit&rdquo; genannt und mit&nbsp; ${\rm Pr}(A, \ B)$&nbsp; bezeichnet.
+
*${\rm Pr}(A ∩ B)$&nbsp; is sometimes called the&nbsp; &raquo;joint probability&laquo;&nbsp; and is denoted by&nbsp; ${\rm Pr}(A, \ B)$.
*${\rm Pr}(A ∩ B)$&nbsp; ist gleich der oberen Schranke, wenn&nbsp; $A$&nbsp; eine&nbsp; [[Theory_of_Stochastic_Signals/Mengentheoretische_Grundlagen#Echte_Teilmenge_.E2.80.93_Unechte_Teilmenge|Teilmenge]]&nbsp;  von&nbsp; $B$&nbsp; ist oder umgekehrt.  
+
 
*Die untere Schranke ergibt sich für die Verbundwahrscheinlichkeit von&nbsp;  [[Theory_of_Stochastic_Signals/Mengentheoretische_Grundlagen#Disjunkte_Mengen|disjunkten Mengen]].
+
*${\rm Pr}(A ∩ B)$&nbsp; is equal to the upper bound if&nbsp; $A$&nbsp; is a&nbsp; [[Theory_of_Stochastic_Signals/Set_Theory_Basics#Proper_subset_.E2.80.93_Improper_subset|$\text{subset}$]]&nbsp;  of&nbsp; $B$&nbsp; or vice versa.
 +
 +
*The lower bound is obtained for the joint probability of&nbsp;  [[Theory_of_Stochastic_Signals/Set_Theory_Basics#Disjoint_sets|&raquo;disjoint sets&laquo;]].}}
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 2:}$&nbsp; Wir betrachten wieder die beiden Ereignisse
+
$\text{Example 2:}$&nbsp; We consider again the experiment&nbsp; &raquo;throwing a die&laquo;.&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
* $A :=$ „die Augenzahl ist größer oder gleich&nbsp; $5$”$ = \{5, 6\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A)= 2/6= 1/3$, und
 
* $B :=$ „die Augenzahl ist geradzahlig”$ = \{2, 4, 6\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (B)= 3/6= 1/2$,
 
  
 +
Consider the two events
 +
* $A :=$&nbsp; &raquo;The outcome is greater than or equal to&nbsp; $5$&laquo;&nbsp;  $ = \{5, 6\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A)= 2/6= 1/3$,&nbsp; 
  
Die Schnittmenge beinhaltet nur ein einziges Element: &nbsp;  $(A ∩ B) = \{ 6 \}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A ∩ B) = 1/6$.
+
* $B :=$&nbsp; &raquo;The outcome is even&laquo;&nbsp; $ = \{2, 4, 6\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (B)= 3/6= 1/2$.  
*Die obere Schranke ergibt sich zu&nbsp; ${\rm Pr} (A ∩ B) \le {\rm Min}\ \big ({\rm Pr} (A), \, {\rm Pr} (B)\big ) = 2/6.$
 
*Die untere Schranke der Schnittmenge ist hier wegen&nbsp; ${\rm Pr} (A) \le 1/2$ &nbsp;und&nbsp; ${\rm Pr} (B) \le 1/2$&nbsp; gleich Null.}}
 
  
==Komplementärmenge==
+
 
 +
The intersection contains only one element: &nbsp;  $(A ∩ B) = \{ 6 \}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A ∩ B) = 1/6$.
 +
*The upper bound is obtained as&nbsp; ${\rm Pr} (A ∩ B) \le {\rm Min}\ \big ({\rm Pr} (A), \, {\rm Pr} (B)\big ) = 2/6.$
 +
 
 +
*The lower bound of the intersection is zero because of&nbsp; ${\rm Pr} (A) \le 1/2$ &nbsp;and&nbsp; ${\rm Pr} (B) \le 1/2$&nbsp;.}}
 +
 
 +
==Complementary set==
 
<br>
 
<br>
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definition:}$&nbsp; Die&nbsp; '''Komplementärmenge'''&nbsp; (englisch:&nbsp; ''Complementary Set'')&nbsp; von&nbsp; $A$&nbsp; wird oft durch eine überstreichende Linie&nbsp; $(\overline{A})$&nbsp; gekennzeichnet.&nbsp; Sie beinhaltet alle die Elemente, die in der Menge&nbsp; $A$&nbsp; nicht enthalten sind, und es gilt für deren Wahrscheinlichkeit:
+
$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''complementary set'''&laquo; of&nbsp; $A$&nbsp; is often denoted by a straight line above the letter&nbsp; $(\overline{A})$&nbsp;.&nbsp; It contains all the elements that are not contained in the set&nbsp; $A$&nbsp; and it holds for their probability:
:$${\rm Pr}(\overline{A}) = 1- {\rm Pr}(A).$$}}
+
[[File:EN_Sto_T_1_2_S4Neu.png| right|frame|Complementary set in the Venn diagram]]
 +
 
 +
:$${\rm Pr}(\overline{A}) = 1- {\rm Pr}(A).$$
  
 +
In the Venn diagram,&nbsp; the set complementary to&nbsp; $A$&nbsp; is shaded.&nbsp;
  
[[File:EN_Sto_T_1_2_S4.png| right|frame|Komplementärmenge im Venndiagramm]]
+
From this diagram,&nbsp; some set-theoretic relationships can be seen:
Im dargestellten  Venndiagramm ist die zu&nbsp; $A$&nbsp; komplementäre Menge schraffiert dargestellt.&nbsp; Aus diesem Schaubild sind einige mengentheoretische Beziehungen zu erkennen:  
+
*The complementary of the complementary of&nbsp; $A$&nbsp; is the set&nbsp; $A$&nbsp; itself:
*Die Komplementärmenge der komplementären Menge von&nbsp; $A$&nbsp; ist die Menge&nbsp; $A$&nbsp; selbst:
+
:$$\overline{\overline{A} } = A.$$
:$$\overline{\overline{A}} = A.$$
+
*The union of a set&nbsp; $A$&nbsp; with its complementary set gives the universal set:
*Die Vereinigungsmenge einer Menge&nbsp; $A$&nbsp; mit ihrer Komplentärmenge ergibt die Grundmenge:
 
 
:$${\rm Pr}(A \cup \overline{A}) = {\rm Pr}(G) = \rm 1.$$
 
:$${\rm Pr}(A \cup \overline{A}) = {\rm Pr}(G) = \rm 1.$$
*Die Schnittmenge von $A$ mit ihrer Komplementärmenge ergibt die leere Menge:
+
*The intersection of&nbsp; $A$&nbsp; with its complementary set gives the empty set:
:$${\rm Pr}(A \cap \overline{A}) = {\rm Pr}({\it \phi}) \rm = 0.$$
+
:$${\rm Pr}(A \cap \overline{A}) = {\rm Pr}({\it \phi}) \rm = 0.$$}}
 +
 
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 3:}$&nbsp; Ausgehend von der Menge 
+
$\text{Example 3:}$&nbsp; We consider again the experiment&nbsp; &raquo;throwing a die&raquo;.&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp; $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
* $A :=$ „die Augenzahl ist kleiner als&nbsp; $5$” $= \{1, 2, 3, 4\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A)= 2/3$
 
  
 +
*Starting from the set
 +
:$$A :=\text{&nbsp;&raquo;The outcome is smaller than&nbsp; $5$&laquo;&nbsp;}  = \{1, 2, 3, 4\}\ \  \text{&nbsp; &rArr; &nbsp;} \ \ {\rm Pr} (A)= 2/3,$$
  
lautet die zugehörige Komplentärmenge:
+
*the corresponding complementary set is
* $\overline{A} :=$ „die Augenzahl ist größer oder gleich&nbsp; $5$”$ = \{5, 6\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (\overline{A})= 1 - {\rm Pr} (A) = 1 - 2/3 = 1/3.$}}
+
:$$\overline{A} :=\text{&nbsp; &raquo;The outcome is greater than or equal to&nbsp; $5$&laquo;}&nbsp;  = \{5, 6\} \ \ \text{&nbsp; &rArr; &nbsp;}\ \  {\rm Pr} (\overline{A})= 1 - {\rm Pr} (A) = 1 - 2/3 = 1/3.$$}}
  
==Echte Teilmenge &ndash; Unechte Teilmenge==
+
==Proper subset &ndash; Improper subset==
 
<br>
 
<br>
[[File:EN_Sto_T_1_2_S5.png | right|frame| Teilmengen im Venndiagramm]]
+
{{BlaueBox|TEXT=
{{BlaueBox|TEXT=  
+
[[File:EN_Sto_T_1_2_S5.png | right|frame| Proper subset in the Venn diagram]]   
$\text{Definitionen:}$&nbsp; Man nennt&nbsp; $A$&nbsp; eine&nbsp; '''echte Teilmenge'''&nbsp; von&nbsp; $B$&nbsp; (englisch:&nbsp; ''Strict Subset'')&nbsp; und schreibt hierfür&nbsp; $A ⊂ B$,  
+
$\text{Definitions:}$&nbsp;  
*wenn alle Elemente von&nbsp; $A$&nbsp; auch in&nbsp; $B$&nbsp; enthalten sind,  
+
 
*aber nicht gleichzeitig alle Elemente von&nbsp; $B$&nbsp; auch in&nbsp; $A$.  
+
'''(1)'''&nbsp; One calls&nbsp; $A$&nbsp; a&nbsp; &raquo;'''proper subset'''&laquo;&nbsp; of&nbsp; $B$&nbsp; and writes for this relationship&nbsp; $A ⊂ B$,  
 +
*if all elements of&nbsp; $A$&nbsp; are also contained in&nbsp; $B$,
 +
 
 +
*but not all elements of&nbsp; $B$&nbsp; are contained in&nbsp; $A$.  
  
  
In diesem Fall gilt für die Wahrscheinlichkeiten:
+
In this case,&nbsp; for the probabilities hold:
 
:$${\rm Pr}(A)  <  {\rm Pr}(B).$$
 
:$${\rm Pr}(A)  <  {\rm Pr}(B).$$
  
Diese mengentheoretische Relation wird durch das skizzierte Venndiagramm veranschaulicht.
+
This set-theoretic relationship is illustrated by the sketched Venn diagram on the right.
  
Dagegen bezeichnet man&nbsp; $A$&nbsp; als eine&nbsp; '''unechte Teilmenge'''&nbsp; von&nbsp; $B$&nbsp; und verwendet die Notation
+
 
 +
'''(2)'''&nbsp; On the other hand,&nbsp; $A$&nbsp; is called an&nbsp; &raquo;'''improper subset'''&laquo;&nbsp; of&nbsp; $B$&nbsp; and uses the notation
 
:$$A \subseteq B = (A \subset B) \cup (A = B),$$
 
:$$A \subseteq B = (A \subset B) \cup (A = B),$$
wenn&nbsp; $A$&nbsp; entweder eine echte Teilmenge von&nbsp; $B$&nbsp; ist oder wenn&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; gleiche Mengen sind.
+
if&nbsp; $A$&nbsp; is either a proper subset of&nbsp; $B$&nbsp; or if&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; are equal sets.
  
*Für die Wahrscheinlichkeiten gilt dann die Größenrelation&nbsp;  ${\rm Pr} (A) \le  {\rm Pr} (B)$.
+
*Then applies to the probabilities:&nbsp;  ${\rm Pr} (A) \le  {\rm Pr} (B)$.
*Das Gleichheitszeichen gilt nur für den Sonderfall&nbsp;  $A = B$.}}
 
  
 +
*The equality sign is only valid for the special case&nbsp;  $A = B$.}}
  
Daneben gelten aber auch die beiden als&nbsp; '''Absorptionsgesetze'''&nbsp; bekannten Gleichungen:
+
 
 +
In addition,&nbsp; the two equations known as the&nbsp; &raquo;'''absorption laws'''&laquo;&nbsp; also apply:
 
:$$(A \cap B)  \cup A  =  A ,$$
 
:$$(A \cap B)  \cup A  =  A ,$$
 
:$$(A  \cup B) \cap A  =  A,$$
 
:$$(A  \cup B) \cap A  =  A,$$
  
da die Schnittmenge&nbsp; $A ∩ B$&nbsp; stets eine Teilmenge von&nbsp; $A$&nbsp; ist, aber gleichzeitig auch&nbsp; $A$&nbsp; eine Teilmenge der Vereinigungsmenge&nbsp; $A ∪ B$&nbsp; ist.  
+
*since the intersection&nbsp; $A ∩ B$&nbsp; is always a subset of&nbsp; $A$,&nbsp;  
 +
*but at the same time&nbsp; $A$&nbsp; is also a subset of the union&nbsp; $A ∪ B$.
 +
 
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 4:}$&nbsp; Wir betrachten bei unserem Standardexperiment "Werfen eines Würfels&rdquo; &nbsp; &rArr; &nbsp; $G =  \{1, 2, 3, 4, 5, 6\}$ nun die beiden Ereignisse
+
$\text{Example 4:}$&nbsp; We consider again the experiment&nbsp; &raquo;throwing a die&laquo;.&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
* $A :=$ „die Augenzahl ist ungerade”$ = \{1, 3, 5\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A)= 3/6$, und
 
* $B :=$ „die Augenzahl ist eine Primzahl” $= \{1, 2, 3, 5\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (B)= 4/6$.
 
  
 +
Consider the two events
 +
* $A :=$&nbsp; &raquo;The outcome is odd&laquo; $&nbsp; = \{1, 3, 5\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A)= 3/6$,&nbsp;
  
Man erkennt, dass&nbsp; $A$&nbsp; eine (echte) Teilmenge der Menge&nbsp; $B$&nbsp; ist.&nbsp; Dementsprechend gilt auch&nbsp; ${\rm Pr} (A) <  {\rm Pr} (B).$ }}
+
* $B :=$&nbsp; &raquo;The outcome is a prime number&raquo; $&nbsp; = \{1, 2, 3, 5\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (B)= 4/6$.
  
==Theoreme von de Morgan==
+
 
 +
It can be seen that&nbsp;  $A$&nbsp; is a&nbsp; $($proper$)$ subset&nbsp; of&nbsp; $B$.&nbsp; Accordingly,&nbsp; ${\rm Pr} (A) <  {\rm Pr} (B)$&nbsp; is also true. }}
 +
 
 +
==Theorems of de Morgan==
 
<br>
 
<br>
[[File:EN_Sto_T_1_2_S6.png|frame| Zu den Theoremen von de Morgan | rechts]]
+
In many set-theoretical tasks,&nbsp; the two theorems of&nbsp; [https://en.wikipedia.org/wiki/Augustus_De_Morgan $\text{de Morgan}$]&nbsp;   are extremely useful.&nbsp;
Bei vielen Aufgaben aus der Mengenlehre sind die beiden Theoreme von&nbsp; [https://de.wikipedia.org/wiki/Augustus_De_Morgan de Morgan]&nbsp;  äußerst nützlich. Diese lauten:
+
   
 +
{{BlaueBox|TEXT=
 +
$\text{Theorem of de Morgan:}$
 +
[[File:EN_Sto_T_1_2_S6.png|frame| Zu den Theoremen von de Morgan | About de Morgan's theorems]]
 +
 
 
:$$\overline{A \cup B} = \overline{A} \cap \overline{B},$$
 
:$$\overline{A \cup B} = \overline{A} \cap \overline{B},$$
 
:$$\overline{A \cap B} = \overline{A} \cup \overline{B}.$$
 
:$$\overline{A \cap B} = \overline{A} \cup \overline{B}.$$
  
Diese Gesetzmäßigkeiten sind im Schaubild veranschaulicht:  
+
These regularities are illustrated in the Venn diagram:
*Die Menge&nbsp; $A$&nbsp; ist rot dargestellt und die Menge&nbsp; $B$&nbsp; blau.  
+
#Set&nbsp; $A$&nbsp; is shown in red and set&nbsp; $B$&nbsp; is shown in blue.  
*Die Komplentärmenge&nbsp; $\overline {A}$&nbsp; von&nbsp; $A$&nbsp; ist in horizontaler Richtung schraffiert.  
+
#The complement&nbsp; $\overline {A}$&nbsp; of&nbsp; $A$&nbsp; is hatched in the horizontal direction.
*Die Komplentärmenge&nbsp;  $\overline {B}$&nbsp; von&nbsp; $B$&nbsp; ist in vertikaler Richtung schraffiert.  
+
#The complement&nbsp;  $\overline {B}$&nbsp; of&nbsp; $B$&nbsp; is hatched in the vertical direction.  
*Das Komplement&nbsp; $\overline{A \cup B}$&nbsp; der Vereinigungsmenge&nbsp; ${A \cup B}$&nbsp; ist sowohl horizontal als auch vertikal schraffiert.  
+
#The complement&nbsp; $\overline{A \cup B}$&nbsp; of the union&nbsp; ${A \cup B}$&nbsp; is hatched both horizontally and vertically.  
*Es ist damit gleich der Schnittmenge&nbsp; $\overline{A} \cap \overline{B}$&nbsp; der beiden Komplentärmengen von&nbsp; $A$&nbsp; und&nbsp; $B$:
+
#It is thus equal to the intersection&nbsp; $\overline{A} \cap \overline{B}$&nbsp; of the two complement sets of&nbsp; $A$&nbsp; and&nbsp; $B$:
:$$\overline{A \cup B} = \overline{A} \cap \overline{B}.$$
+
::$$\overline{A \cup B} = \overline{A} \cap \overline{B}.$$}}
  
Auch die zweite Form des de Morgan-Theorems lässt sich mit diesem Venndiagramm grafisch verdeutlichen:
 
  
*Die Schnittmenge&nbsp; $A ∩ B$&nbsp; (im Bild violett dargestellt) ist weder horizontal noch vertikal schraffiert.  
+
The second form of the de Morgan theorem can also be illustrated graphically with the same Venn diagram:
*Das Komplement&nbsp; $\overline{A ∩ B}$&nbsp; der Schnittmenge ist dementsprechend entweder horizontal, vertikal oder in beiden Richtungen schraffiert.  
+
 
*Nach dem zweiten Theorem von de Morgan ist das Komplement der Schnittmenge gleich der Vereinigungsmenge der beiden Komplentärmengen von&nbsp; $A$&nbsp; und&nbsp; $B$:
+
#The intersection&nbsp; $A ∩ B$&nbsp; $($shown in purple in the figure$)$&nbsp; is neither horizontally nor vertically hatched.  
:$$\overline{A \cap B} = \overline{A} \cup \overline{B}.$$
+
#Accordingly, the complement&nbsp; $\overline{A ∩ B}$&nbsp; of the intersection is hatched either horizontally, vertically, or in both directions.
 +
#By de Morgan's second theorem,&nbsp; the complement of the intersection equals the union of the two complementary sets of&nbsp; $A$&nbsp; and&nbsp; $B$:
 +
::$$\overline{A \cap B} = \overline{A} \cup \overline{B}.$$
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 5:}$&nbsp; Wir betrachten die beiden Mengen
+
$\text{Example 5:}$&nbsp; We consider again the experiment&nbsp; &raquo;throwing a die&laquo;.&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
* $A : =$ „die Augenzahl ist ungeradzahlig” $= \{1, 3, 5\}$,
 
* $B : =$  „die Augenzahl ist größer als&nbsp; $2$” $= \{3, 4, 5, 6\}$.  
 
  
 +
We consider the two sets
 +
* $A : =$&nbsp; &raquo;The outcome is odd&laquo;&nbsp; $= \{1, 3, 5\}$,
 +
* $B : =$&nbsp; &raquo;The outcome is greater than&nbsp; $2$&laquo;&nbsp; $= \{3, 4, 5, 6\}$.
  
Daraus folgen die beiden komplementären Mengen
 
* $\overline {A} : =$ „die Augenzahl ist geradzahlig” $= \{2, 4, 6\}$,
 
* $\overline {B} : =$ „die Augenzahl ist kleiner als&nbsp; $3$” $= \{1, 2\}$.
 
  
 +
From this follow the two complementary sets
 +
* $\overline {A} : =$&nbsp; &raquo;The outcome is even&laquo;&nbsp; $= \{2, 4, 6\}$,
 +
* $\overline {B} : =$&nbsp; &raquo;The outcome is smaller than&nbsp; $3$&laquo;&nbsp; $= \{1, 2\}$.
  
Weiter erhält man mit den obigen Theoremen die folgenden Teilmengen:
 
:$$\overline{A \cup B} = \overline{A} \cap \overline{B} = \{2\}\hspace{0.5 cm}\rm und \hspace{0.5cm} \overline{\it A \cap \it B} =\overline{\it A} \cup \overline{\it B} = \{1,2,4,6\}.$$}}
 
  
==Disjunkte Mengen==
+
Further,&nbsp; using the above theorems,&nbsp; we obtain the following sets:
 +
:$$\overline{A \cup B} = \overline{A} \cap \overline{B} = \{2\},$$
 +
:$$\overline{\it A \cap \it B} =\overline{\it A} \cup \overline{\it B} = \{1,2,4,6\}.$$}}
 +
 
 +
==Disjoint sets==
 
<br>
 
<br>
{{BlaueBox|TEXT=
+
{{BlaueBox|TEXT=
$\text{Definition:}$&nbsp; Zwei Mengen&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; nennt man&nbsp; '''disjunkt'''&nbsp; (englisch:&nbsp; ''disjoint'')&nbsp; oder&nbsp; '''miteinander unvereinbar''',  
+
[[File:EN_Sto_T_1_2_S7.png |frame| Disjunkte Mengen im Venndiagramm | Disjoint sets in the Venn diagram]]
*wenn es kein einziges Element gibt,  
+
 
*das sowohl in&nbsp; $A$&nbsp; als auch in&nbsp; $B$&nbsp; enthalten ist.}}
+
$\text{Definition:}$&nbsp; Two sets&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; are called&nbsp; &raquo;'''disjoint'''&laquo; or&nbsp; &raquo;'''incompatible'''&laquo;,
 +
 +
*if there is no single element,
 +
 +
*that is contained in both&nbsp; $A$&nbsp; and&nbsp; $B$.
  
  
[[File:EN_Sto_T_1_2_S7.png |frame| Disjunkte Mengen im Venndiagramm | rechts]]
+
The diagram shows two disjoint sets&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; in the Venn diagram.
Das Schaubild zeigt zwei disjunkte Mengen&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; im Venndiagramm.  
 
  
In diesem Sonderfall gelten die folgenden Aussagen:  
+
In this special case,&nbsp; the following statements hold:
*Die Schnittmenge zweier disjunkter Mengen&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; ergibt stets die leere Menge:
+
 +
*The intersection of two disjoint sets&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; always yields the empty set:
 
:$${\rm Pr}(A \cap B) =  {\rm Pr}(\phi) = \rm 0.$$
 
:$${\rm Pr}(A \cap B) =  {\rm Pr}(\phi) = \rm 0.$$
*Die Wahrscheinlichkeit der Vereinigungsmenge zweier disjunkter Mengen&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; ist immer gleich der Summe der beiden Einzelwahrscheinlichkeiten:
+
*The probability of the union set of two disjoint sets&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; is always equal to the sum of the two individual probabilities:
:$${\rm Pr}( A \cup B) =  {\rm Pr}( A) + {\rm Pr}(B).$$
+
:$${\rm Pr}( A \cup B) =  {\rm Pr}( A) + {\rm Pr}(B).$$}}
 
<br clear=all>
 
<br clear=all>
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 6:}$&nbsp; Bei unserem Standardexperiment sind die beiden Mengen
+
$\text{Example 6:}$&nbsp; We consider again the experiment&nbsp; &raquo;throwing a die&laquo;.&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
* $A :=$  „die Augenzahl ist kleiner als&nbsp; $3$”$ = \{1, 2\}$  &nbsp; ⇒  &nbsp; ${\rm Pr}( A) = 2/6$, und
 
* $B :=$  „die Augenzahl ist größer als&nbsp; $3$” $ = \{4, 5,6\}$ &nbsp; ⇒  &nbsp;  ${\rm Pr}( B) = 3/6$
 
  
 +
In our standard experiment,&nbsp; the two sets are now
 +
* $A :=$&nbsp;  &raquo;The outcome is smaller than&nbsp; $3$ &laquo; $ = \{1, 2\}$  &nbsp; ⇒  &nbsp; ${\rm Pr}( A) = 2/6$,
 +
 
 +
* $B :=$&nbsp;  &raquo;The outcome is greater than&nbsp; $3$ &laquo; $ = \{4, 5,6\}$  &nbsp; ⇒  &nbsp;  ${\rm Pr}( B) = 3/6$
  
zueinander disjunkt, da&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; kein einziges gemeinsames Element beinhalten.
 
*Die Schnittmenge ergibt die leere Menge:&nbsp; ${A \cap B} = \phi$.
 
*Die Wahrscheinlichkeit der Vereinigungsmenge&nbsp; ${A \cup B}  = \{1, 2, 4, 5, 6\}$&nbsp; ist gleich&nbsp; ${\rm Pr}( A) + {\rm Pr}(B) = 5/6.$}}
 
  
==Additionstheorem==
+
disjoint to each other,&nbsp; since&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; do not contain a single common element.
 +
#The intersection yields the empty set:&nbsp; ${A \cap B} = \phi$.
 +
#The probability of the union set&nbsp; ${A \cup B}  = \{1, 2, 4, 5, 6\}$&nbsp; is equal to&nbsp; ${\rm Pr}( A) + {\rm Pr}(B) = 5/6.$}}
 +
 
 +
==Addition rule==
 
<br>
 
<br>
Nur bei disjunkten Mengen&nbsp; $A$&nbsp; und&nbsp; $B$&nbsp; gilt für die Wahrscheinlichkeit der Vereinigungsmenge der Zusammenhang&nbsp; ${\rm Pr}( A \cup B) =  {\rm Pr}( A) + {\rm Pr}(B)$.&nbsp; Wie errechnet sich diese Wahrscheinlichkeit aber bei allgemeinen, nicht notwendigerweise disjunkten Ereignissen?  
+
Only for disjoint sets&nbsp; $A$&nbsp; and&nbsp; $B$,&nbsp; the relation&nbsp; ${\rm Pr}( A \cup B) =  {\rm Pr}( A) + {\rm Pr}(B)$&nbsp; holds for the probability of the union set.&nbsp; But how is this probability calculated for general events that are not necessarily disjoint?  
  
[[File:EN_Sto_T_1_2_S8.png | right|frame| Zum Additionstheorem der Wahrscheinlichkeitsrechnung]]
+
[[File:EN_Sto_T_1_2_S8.png | right|frame| &raquo;Addition rule&laquo;&nbsp; of probability calculus]]
Betrachten Sie das rechte  Venndiagramm mit der violett dargestellten Schnittmenge&nbsp; $A ∩ B$.
+
Consider the right-hand Venn diagram with the intersection&nbsp; $A ∩ B$&nbsp; shown in purple:
*Die rote Menge beinhaltet alle Elemente, die zu&nbsp; $A$&nbsp; gehören, aber nicht zu&nbsp; $B$.  
+
#The red set contains all the elements that belong to&nbsp; $A$,&nbsp; but not to&nbsp; $B$.  
*Die Elemente von&nbsp; $B$, die nicht gleichzeitig in&nbsp; $A$&nbsp; enthalten sind, sind blau dargestellt.  
+
#The elements of&nbsp; $B$, that are not simultaneously contained in&nbsp; $A$&nbsp; are shown in blue.  
*Alle roten, blauen und violetten Flächen zusammen ergeben die Vereinigungsmenge&nbsp; $A ∪ B$.
+
#All red,&nbsp; blue,&nbsp; and purple surfaces together make up the union set&nbsp; $A ∪ B$.
  
  
Aus dieser mengentheoretischen Darstellung erkennt man folgende Zusammenhänge:
+
From this set-theoretical representation,&nbsp; one can see the following relationships:
 
:$${\rm Pr}(A) \hspace{0.8cm}= {\rm Pr}(A \cap B)  + {\rm Pr}(A \cap \overline{B}),$$
 
:$${\rm Pr}(A) \hspace{0.8cm}= {\rm Pr}(A \cap B)  + {\rm Pr}(A \cap \overline{B}),$$
 
:$${\rm Pr}(B) \hspace{0.8cm}= {\rm Pr}(A \cap B) \rm +{\rm Pr}(\overline{A} \cap {B}),$$
 
:$${\rm Pr}(B) \hspace{0.8cm}= {\rm Pr}(A \cap B) \rm +{\rm Pr}(\overline{A} \cap {B}),$$
:$${\rm Pr}(A \cup B) ={\rm Pr}(A \cap B) +{\rm Pr} ({A} \cap \overline{B}) \rm + {\rm Pr}(\overline{A} \cap {B}).$$
+
:$${\rm Pr}(A \cup B) ={\rm Pr}(A \cap B) +{\rm Pr} ({A} \cap \overline{B}) + {\rm Pr}(\overline{A} \cap {B}).$$
  
Addiert man die ersten beiden Gleichungen und subtrahiert davon die dritte, so erhält man:
+
Adding the first two equations and subtracting from them the third,&nbsp; we get:
:$${\rm Pr}(A) \rm +{\rm Pr}(B) -{\rm Pr}(A \cup B) = {\rm Pr}(A \cap B).$$
+
:$${\rm Pr}(A) +{\rm Pr}(B) -{\rm Pr}(A \cup B) = {\rm Pr}(A \cap B).$$
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definition:}$&nbsp; Durch Umstellen dieser Gleichung kommt man zum sogenannten&nbsp; '''Additionstheorem'''&nbsp; (englisch:&nbsp; ''Addition Rule'')&nbsp; für zwei beliebige, nicht notwendigerweise disjunkte Ereignisse:
+
$\text{Definition:}$&nbsp; By rearranging this equation,&nbsp; one arrives at the so-called&nbsp; &raquo;'''addition rule'''&laquo;&nbsp; for any two,&nbsp; not necessarily disjoint events:
 
:$${\rm Pr}(A \cup B) = {\rm Pr}(A) + {\rm Pr}(B) - {\rm Pr}(A \cap B).$$}}
 
:$${\rm Pr}(A \cup B) = {\rm Pr}(A) + {\rm Pr}(B) - {\rm Pr}(A \cap B).$$}}
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 7:}$&nbsp; Wir betrachten die beiden Mengen
+
$\text{Example 7:}$&nbsp; We consider again the experiment&nbsp; &raquo;throwing a die&laquo;.&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
* $A :=$ „die Augenzahl ist ungeradzahlig” $= \{1, 3, 5\}$  &nbsp; ⇒ &nbsp;  ${\rm Pr}(A) = 3/6$, und
+
 
* $B :=$ „die Augenzahl ist größer als&nbsp; $2$$ = \{3, 4, 5, 6\}$  &nbsp; ⇒ &nbsp;  ${\rm Pr}(B) = 4/6$.  
+
We consider the two sets
 +
* $A :=$&nbsp; &raquo;The outcome is odd &laquo; $= \{1, 3, 5\}$  &nbsp; ⇒ &nbsp;  ${\rm Pr}(A) = 3/6$,
 +
 
 +
* $B :=$&nbsp; &raquo;The outcome is greater than&nbsp; $2$ &laquo; $ = \{3, 4, 5, 6\}$  &nbsp; ⇒ &nbsp;  ${\rm Pr}(B) = 4/6$.  
  
  
Damit ergeben sich für die Wahrscheinlichkeiten
+
This gives the following probabilities
*der Vereinigungsmenge &nbsp; ⇒ &nbsp;  ${\rm Pr}(A ∪ B) = 5/6$, und
+
*of the union &nbsp; ⇒ &nbsp;  ${\rm Pr}(A ∪ B) = 5/6$,&nbsp; and
*der Schnittmenge &nbsp; ⇒ &nbsp;  ${\rm Pr}(A  ∩ B) = 2/6$.  
+
 +
*of the intersection &nbsp; ⇒ &nbsp;  ${\rm Pr}(A  ∩ B) = 2/6$.  
  
  
Die Zahlenwerte zeigen die Gültigkeit des Additionstheorems: &nbsp; $5/6 = 3/6 + 4/6 − 2/6$.}}
+
The numerical values show the validity of the addition rule: &nbsp;  
 +
:$$5/6 = 3/6 + 4/6 − 2/6.$$}}
  
==Vollständiges System==
+
==Complete system==
 
<br>
 
<br>
Im letzten Abschnitt zu diesem Kapitel betrachten wir wieder mehr als zwei mögliche Ereignisse, nämlich allgemein&nbsp; $I$.&nbsp; Diese Ereignisse werden im Folgenden mit&nbsp; $A_i$&nbsp; bezeichnet, und es gilt für den Laufindex: &nbsp; $1 ≤ i ≤ I$.
+
In the last section to this chapter,&nbsp; we consider again more than two possible events, namely, in general,&nbsp; $I$.&nbsp; These events will be denoted by&nbsp; $A_i$ &nbsp; &rArr; &nbsp; the running index $i$ can be in the range&nbsp; $1 ≤ i ≤ I$.
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definition:}$&nbsp; Eine Konstellation mit den Ereignissen&nbsp; $A_1, \hspace{0.1cm}\text{...}\hspace{0.1cm} , A_i,  \hspace{0.1cm}\text{...}\hspace{0.1cm}  , A_I$&nbsp; bezeichnet man dann und nur dann als ein&nbsp; '''vollständiges System''', wenn die beiden folgenden Bedingungen erfüllt sind:
+
$\text{Definition:}$&nbsp; A constellation with events&nbsp; $A_1, \hspace{0.1cm}\text{...}\hspace{0.1cm} , A_i,  \hspace{0.1cm}\text{...}\hspace{0.1cm}  , A_I$&nbsp; is called a&nbsp; &raquo;'''complete system'''&laquo;,&nbsp; if and only if the following two conditions are satisfied:
 
   
 
   
'''(1)''' &nbsp; Alle Ereignisse sind paarweise disjunkt:
+
'''(1)''' &nbsp; All events are pairwise disjoint:
:$$A_i \cap A_j = \it \phi \hspace{0.15cm}\rm f\ddot{u}r\hspace{0.15cm}alle\hspace{0.15cm}\it i \ne j.$$
+
:$$A_i \cap A_j = \it \phi \hspace{0.25cm}\rm for\hspace{0.15cm}all\hspace{0.25cm}\it i \ne j.$$
'''(2)''' &nbsp; Die Vereinigung aller Ereignismengen ergibt die Grundmenge:
+
'''(2)''' &nbsp; The union of all event sets gives the universal set:
 
:$$\bigcup_{i=1}^{I} A_i = G.$$}}
 
:$$\bigcup_{i=1}^{I} A_i = G.$$}}
  
  
Aufgrund dieser beiden Voraussetzungen gilt dann für die Summe aller Wahrscheinlichkeiten:
+
Given these two assumptions, the sum of all probabilities is then:
 
:$$\sum_{i =1}^{  I} {\rm Pr}(A_i) = 1.$$
 
:$$\sum_{i =1}^{  I} {\rm Pr}(A_i) = 1.$$
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 8:}$&nbsp; Die Ereignismengen&nbsp; $A_1 := \{1, 5\}$&nbsp; und&nbsp; $A_2 := \{2, 3\}$&nbsp; ergeben beim Zufallsexperiment "Werfen eines Würfels&rdquo; zusammen mit der Menge&nbsp; $A_3 := \{4, 6\}$&nbsp; ein vollständiges System, nicht jedoch beim Experiment "Werfen einer Roulettekugel&rdquo;.}}
+
$\text{Example 8:}$&nbsp;  
 +
*The sets&nbsp; $A_1 := \{1, 5\}$&nbsp; and&nbsp; $A_2 := \{2, 3\}$&nbsp; together with the set&nbsp; $A_3 := \{4, 6\}$&nbsp; result in a complete system in the random experiment&nbsp; &raquo;throwing a die&laquo;,
 +
 
 +
* but not in the experiment&nbsp; &raquo;throwing a roulette ball&laquo;.}}
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 9:}$&nbsp; Ein weiteres Beispiel für ein vollständiges System ist die diskrete Zufallsgröße&nbsp; $X = \{ x_1, x_2, \hspace{0.1cm}\text{...}\hspace{0.1cm} , x_I\}$&nbsp; mit den Auftrittswahrscheinlichkeiten entsprechend der folgenden&nbsp; [[Information_Theory/Einige_Vorbemerkungen_zu_zweidimensionalen_Zufallsgrößen#Wahrscheinlichkeitsfunktion_und_Wahrscheinlichkeitsdichtefunktion|Wahrscheinlichkeitsfunktion]]:
+
$\text{Example 9:}$&nbsp; Another example of a complete system is the discrete random variable&nbsp; $X = \{ x_1, x_2, \hspace{0.1cm}\text{...}\hspace{0.1cm} , x_I\}$&nbsp; with the likelihood corresponding to the following&nbsp; [[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Probability_mass_function_and_probability_density_function|&raquo;probability mass function&laquo;]]&nbsp; $\rm (PMF)$:
:$$P_X(X) = \big [ \hspace{0.1cm} P_X(x_1), P_X(x_2), \hspace{0.05cm}\text{...}\hspace{0.15cm}, P_X(x_I) \hspace{0.05cm} \big ] = \big [ \hspace{0.1cm} p_1, p_2, \hspace{0.05cm}\text{...} \hspace{0.15cm}, p_I \hspace{0.05cm} \big ] \hspace{0.05cm}$$
+
:$$P_X(X) = \big [ \hspace{0.1cm} P_X(x_1),\ P_X(x_2), \hspace{0.05cm}\text{...}\hspace{0.15cm},\ P_X(x_I) \hspace{0.05cm} \big ] = \big [ \hspace{0.1cm} p_1, p_2, \hspace{0.05cm}\text{...} \hspace{0.15cm}, p_I \hspace{0.05cm} \big ] \hspace{0.05cm}$$
 
:$$\Rightarrow \hspace{0.3cm} p_1 = P_X(x_1) = {\rm Pr}(X=x_1) \hspace{0.05cm},  
 
:$$\Rightarrow \hspace{0.3cm} p_1 = P_X(x_1) = {\rm Pr}(X=x_1) \hspace{0.05cm},  
 
\hspace{0.2cm}p_2 =  {\rm Pr}(X=x_2) \hspace{0.05cm},\hspace{0.05cm}\text{...}\hspace{0.15cm},\hspace{0.2cm} p_I = {\rm Pr}(X=x_I) \hspace{0.05cm}.$$
 
\hspace{0.2cm}p_2 =  {\rm Pr}(X=x_2) \hspace{0.05cm},\hspace{0.05cm}\text{...}\hspace{0.15cm},\hspace{0.2cm} p_I = {\rm Pr}(X=x_I) \hspace{0.05cm}.$$
  
Die möglichen Ergebnisse&nbsp; $x_i$&nbsp; der Zufallsgröße&nbsp; $X$&nbsp; sind paarweise zueinander disjunkt und die Summe aller Auftrittswahrscheinlichkeiten&nbsp;  $p_1 + p_2 + \hspace{0.1cm}\text{...}\hspace{0.1cm} + \hspace{0.05cm} p_I$&nbsp;  liefert grundsätzlich das Ergebnis&nbsp; $1$.}}
+
*The possible outcomes&nbsp; $x_i$&nbsp; of the random variable&nbsp; $X$&nbsp; are pairwise disjoint to each other.
 +
 
 +
*The sum of all likelihoods&nbsp;  $p_1 + p_2 + \hspace{0.1cm}\text{...}\hspace{0.1cm} + \hspace{0.05cm} p_I$&nbsp;  always yields the result&nbsp; $1$.}}
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 10:}$&nbsp; Es gelte&nbsp; $X= \{0, 1, 2 \}$&nbsp; und&nbsp; $P_X (X) = \big[0.2, \ 0.5, \ 0.3\big]$. Dann gilt:
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$\text{Example 10:}$&nbsp; Let&nbsp; $X= \{0, 1, 2 \}$&nbsp; and&nbsp; $P_X (X) = \big[0.2, \ 0.5, \ 0.3\big]$. Then holds:
 
:$${\rm Pr}(X=0) = 0.2 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X=1) = 0.5 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X=2) = 0.3 \hspace{0.05cm}.$$
 
:$${\rm Pr}(X=0) = 0.2 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X=1) = 0.5 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X=2) = 0.3 \hspace{0.05cm}.$$
  
Bei der Zufallsgröße&nbsp; $X = \{1, \pi, {\rm e} \}$&nbsp; und gleichem&nbsp; $P_X(X) = \big[0.2, \ 0.5, \ 0.3\big]$&nbsp; lauten die Zuordnungen:
+
With random variable&nbsp; $X = \{1, \pi, {\rm e} \}$&nbsp; and the same&nbsp; $P_X(X) = \big[0.2, \ 0.5, \ 0.3\big]$&nbsp; the assignments are:
 
:$${\rm Pr}(X=1) = 0.2 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X=\pi) = 0.5 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X={\rm e}) = 0.3 \hspace{0.05cm}.$$
 
:$${\rm Pr}(X=1) = 0.2 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X=\pi) = 0.5 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X={\rm e}) = 0.3 \hspace{0.05cm}.$$
  
 +
$\text{Hints:}$
 +
*The&nbsp; [[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Probability_mass_function_and_probability_density_function|&raquo;probability mass function&laquo;]]&nbsp; $P_X(X)$&nbsp; only makes statements about probabilities,&nbsp; not about the set of values&nbsp;  $\{x_1, x_2, \hspace{0.1cm}\text{...}\hspace{0.1cm} , x_I\}$&nbsp; of the random variable&nbsp; $X$.
 +
 +
*This additional information is provided by the&nbsp; [[Theory_of_Stochastic_Signals/Probability_Density_Function_(PDF)#Definition_of_the_probability_density_function|&raquo;probability density function&laquo;]]&nbsp; $\rm (PDF)$.}}
  
''Hinweise:''
+
==Exercises for the chapter==
*Die&nbsp; [[Information_Theory/Einige_Vorbemerkungen_zu_zweidimensionalen_Zufallsgrößen#Wahrscheinlichkeitsfunktion_und_Wahrscheinlichkeitsdichtefunktion|Wahrscheinlichkeitsfunktion]]&nbsp; $P_X(X)$&nbsp; macht nur Aussagen über die Wahrscheinlichkeiten, nicht über den Wertevorrat&nbsp;  $\{x_1, x_2, \hspace{0.1cm}\text{...}\hspace{0.1cm} , x_I\}$&nbsp; der Zufallsgröße&nbsp; $X$.
 
*Diese zusätzliche Information liefert die&nbsp; [[Theory_of_Stochastic_Signals/Wahrscheinlichkeitsdichtefunktion_(WDF)#Definition_der_Wahrscheinlichkeitsdichtefunktion|Wahrscheinlichkeitsdichtefunktion]]&nbsp; (WDF).}}
 
 
 
==Aufgaben zum Kapitel==
 
 
<br>
 
<br>
[[Aufgaben:1.2 Schaltlogik (D/B-Wandler)|Aufgabe 1.2: Schaltlogik (D/B-Wandler)]]
+
[[Aufgaben:Exercise_1.2:_Switching_Logic_(D/B_Converter)|Exercise 1.2: Switching Logic (D/B Converter)]]
  
[[Aufgaben:1.2Z_Ziffernmengen|Aufgabe 1.2Z: Ziffernmengen]]
+
[[Aufgaben:Exercise_1.2Z:_Sets_of_Digits|Exercise 1.2Z: Sets of Digits]]
  
[[Aufgaben:1.3 Fiktive_Uni_Irgenwo|Aufgabe 1.3: Fiktive Uni Irgenwo]]
+
[[Aufgaben:Exercise_1.3:_Fictional_University_Somewhere|Exercise 1.3: Fictional University Somewhere]]
  
[[Aufgaben:Aufgabe_1.3Z:_Gewinnen_mit_Roulette%3F|Aufgabe 1.3Z: Gewinnen mit Roulette?]]
+
[[Aufgaben:Exercise_1.3Z:_Winning_with_Roulette%3F|Exercise 1.3Z: Winning with Roulette?]]
  
  
 
{{Display}}
 
{{Display}}

Latest revision as of 15:41, 5 February 2024

Venn diagram, universal and empty set


In later chapters,  we will sometimes refer to  »set theory« .  Therefore,  the most important basics and definitions of this discipline will be briefly summarized here.  The topic is also covered in the  $($German language$)$  learning video  »Mengentheoretische Begriffe und Gesetzmäßigkeiten«   ⇒   »Set Theory – Terms and Regularities«.

Set representation in the Venn diagram

An important tool of set theory is the  »Venn diagram«  according to the graph:

  • Applied to probability theory,  the events  $A_i$  are represented here as areas.  For a simpler description we do not denote the events here with  $A_1$,  $A_2$  and  $A_3$,  but with  $A$,  $B$  and  $C$ in contrast to the last chapter. 
  • The total area corresponds to the  »universal set«  $($or short:  »universe«$)$  $G$.  The universe  $G$  contains all possible outcomes and stands for the  »certain event«,  which by definition occurs with probability »one«:   ${\rm Pr}(G) = 1$.  For example,  in the random experiment  »Throwing a die«,  the probability for the event  »The number of eyes is less than or equal to 6«  is identical to one.
  • In contrast,  the  »empty set«  $ϕ$  does not contain a single element.  In terms of events,  the empty set specifies the  »impossible event«  with probability  ${\rm Pr}(ϕ) = 0$  an.  For example,  in the experiment  »Throwing a die«,  the probability for the event  »The number of eyes is greater than 6«  is identically zero.


It should be noted that not every event  $A$  with  ${\rm Pr}(A) = 0$  can really never occur:

  • Thus,  the probability of the event  »the noise value  $n$  is identical to zero»  is vanishingly small and it applies  ${\rm Pr}(n \equiv 0) = 0$,  if  $n$  is described by a continuous–valued  $($Gaussian$)$  random variable.
  • Nevertheless,  it is of course possible  $($although extremely unlikely$)$  that at some points the exact noise value  $n = 0$  will also occur.

Union set


Some set-theoretical relations are explained now on the basis of the Venn diagram.

$\text{Definition:}$  The  »union set«  $C$  of two sets  $A$  and  $B$  contains all the elements that are contained either in set  $A$  or in set  $B$  or in both. 

Union set in the Venn diagram
  • This relationship is expressed as the following formula:
$$\ C = A \cup B.$$
  • Using the diagram, it is easy to see the following laws of set theory:
$$A \cup \it \phi = A \rm \hspace{3.6cm}(union \hspace{0.15cm}with \hspace{0.15cm}the \hspace{0.15cm}empty \hspace{0.15cm}set),$$
$$A\cup G = G \rm \hspace{3.6cm}(union \hspace{0.15cm}with \hspace{0.15cm}the \hspace{0.15cm}universe),$$
$$A\cup A = A \hspace{3.6cm}(\rm tautology),$$
$$A\cup B = B\cup A \hspace{2.75cm}(\rm commutative \hspace{0.15cm}property),$$
$$(A\cup B)\cup C = A\cup (B\cup C) \hspace{0.45cm}(\rm associative \hspace{0.15cm}property).$$
  • If nothing else is known about the event sets  $A$  and  $B$  then only a lower bound and an upper bound can be given for the probability of the union set:
$${\rm Max}\big({\rm Pr} (A), \ {\rm Pr} (B)\big) \le {\rm Pr} (A \cup B) \le {\rm Pr} (A)+{\rm Pr} (B).$$
  • The probability of the union set is equal to the lower bound if  $A$  is a  $\text{subset}$  of  $B$  or vice versa.


$\text{Example 1:}$  We consider again the experiment  »throwing a die«.  The possible outcomes  $($number of points$)$  are thus  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.

Consider the two events

  • $A :=$  »The outcome is greater than or equal to  $5$ «  $ = \{5, 6\}$   ⇒   ${\rm Pr} (A)= 2/6= 1/3$,
  • $B :=$  »The outcome is even «  $= \{2, 4, 6\}$   ⇒   ${\rm Pr} (B)= 3/6= 1/2$,


then the union set contains four elements:   $(A \cup B) = \{2, 4, 5, 6 \}$   ⇒   ${\rm Pr} (A \cup B) = 4/6 = 2/3$.

  • For the lower bound:   ${\rm Pr} (A \cup B) \ge {\rm Max}\big({\rm Pr} (A),\ {\rm Pr} (B)\big ) = 3/6.$
  • For the upper bound:   $ {\rm Pr} (A \cup B) \le {\rm Pr} (A)+{\rm Pr} (B) = 5/6.$

Intersection set


Another important set-theoretic relation is the intersection.

$\text{Definition:}$  The  »intersection set«  $C$  of two sets  $A$  and  $B$  contains all those elements which are contained in both the set  $A$  and the set  $B$.

Intersection set in the Venn diagram
  • This relationship is expressed as the following formula:
$$C = A \cap B.$$
  • In the diagram,  the intersection is shown in purple.  Analog to the union set,  the following regularities apply here:
$$A \cap \it \phi = \it \phi \rm \hspace{3.75cm}(intersection \hspace{0.15cm}with \hspace{0.15cm}the \hspace{0.15cm}empty \hspace{0.15cm}set),$$
$$A \cap G = A \rm \hspace{3.6cm}(intersection \hspace{0.15cm}with \hspace{0.15cm}the \hspace{0.15cm}universe),$$
$$A\cap A = A \rm \hspace{3.6cm}(tautology),$$
$$A\cap B = B\cap A \rm \hspace{2.75cm}(commutative \hspace{0.15cm}property),$$
$$(A\cap B)\cap C = A\cap (B\cap C) \rm \hspace{0.45cm}(associative \hspace{0.15cm}property).$$
  • If nothing else is known about  $A$  and  $B$,  then no statement can be made for the probability of the intersection.
  • However,  if  ${\rm Pr} (A) \le 1/2$  and at the same time  ${\rm Pr} (B) \le 1/2$ hold,  then a lower and an upper bound can be given:
$$0 \le {\rm Pr} (A ∩ B) \le {\rm Min}\ \big({\rm Pr} (A),\ {\rm Pr} (B)\big ).$$
  • ${\rm Pr}(A ∩ B)$  is sometimes called the  »joint probability«  and is denoted by  ${\rm Pr}(A, \ B)$.
  • ${\rm Pr}(A ∩ B)$  is equal to the upper bound if  $A$  is a  $\text{subset}$  of  $B$  or vice versa.


$\text{Example 2:}$  We consider again the experiment  »throwing a die«.  The possible outcomes  (number of points)  are thus  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.

Consider the two events

  • $A :=$  »The outcome is greater than or equal to  $5$«  $ = \{5, 6\}$   ⇒   ${\rm Pr} (A)= 2/6= 1/3$, 
  • $B :=$  »The outcome is even«  $ = \{2, 4, 6\}$   ⇒   ${\rm Pr} (B)= 3/6= 1/2$.


The intersection contains only one element:   $(A ∩ B) = \{ 6 \}$   ⇒   ${\rm Pr} (A ∩ B) = 1/6$.

  • The upper bound is obtained as  ${\rm Pr} (A ∩ B) \le {\rm Min}\ \big ({\rm Pr} (A), \, {\rm Pr} (B)\big ) = 2/6.$
  • The lower bound of the intersection is zero because of  ${\rm Pr} (A) \le 1/2$  and  ${\rm Pr} (B) \le 1/2$ .

Complementary set


$\text{Definition:}$  The  »complementary set« of  $A$  is often denoted by a straight line above the letter  $(\overline{A})$ .  It contains all the elements that are not contained in the set  $A$  and it holds for their probability:

Complementary set in the Venn diagram
$${\rm Pr}(\overline{A}) = 1- {\rm Pr}(A).$$

In the Venn diagram,  the set complementary to  $A$  is shaded. 

From this diagram,  some set-theoretic relationships can be seen:

  • The complementary of the complementary of  $A$  is the set  $A$  itself:
$$\overline{\overline{A} } = A.$$
  • The union of a set  $A$  with its complementary set gives the universal set:
$${\rm Pr}(A \cup \overline{A}) = {\rm Pr}(G) = \rm 1.$$
  • The intersection of  $A$  with its complementary set gives the empty set:
$${\rm Pr}(A \cap \overline{A}) = {\rm Pr}({\it \phi}) \rm = 0.$$


$\text{Example 3:}$  We consider again the experiment  »throwing a die».  The possible outcomes  (number of points)  are thus  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.

  • Starting from the set
$$A :=\text{ »The outcome is smaller than  $5$« } = \{1, 2, 3, 4\}\ \ \text{  ⇒  } \ \ {\rm Pr} (A)= 2/3,$$
  • the corresponding complementary set is
$$\overline{A} :=\text{  »The outcome is greater than or equal to  $5$«}  = \{5, 6\} \ \ \text{  ⇒  }\ \ {\rm Pr} (\overline{A})= 1 - {\rm Pr} (A) = 1 - 2/3 = 1/3.$$

Proper subset – Improper subset


Proper subset in the Venn diagram

$\text{Definitions:}$ 

(1)  One calls  $A$  a  »proper subset«  of  $B$  and writes for this relationship  $A ⊂ B$,

  • if all elements of  $A$  are also contained in  $B$,
  • but not all elements of  $B$  are contained in  $A$.


In this case,  for the probabilities hold:

$${\rm Pr}(A) < {\rm Pr}(B).$$

This set-theoretic relationship is illustrated by the sketched Venn diagram on the right.


(2)  On the other hand,  $A$  is called an  »improper subset«  of  $B$  and uses the notation

$$A \subseteq B = (A \subset B) \cup (A = B),$$

if  $A$  is either a proper subset of  $B$  or if  $A$  and  $B$  are equal sets.

  • Then applies to the probabilities:  ${\rm Pr} (A) \le {\rm Pr} (B)$.
  • The equality sign is only valid for the special case  $A = B$.


In addition,  the two equations known as the  »absorption laws«  also apply:

$$(A \cap B) \cup A = A ,$$
$$(A \cup B) \cap A = A,$$
  • since the intersection  $A ∩ B$  is always a subset of  $A$, 
  • but at the same time  $A$  is also a subset of the union  $A ∪ B$.


$\text{Example 4:}$  We consider again the experiment  »throwing a die«.  The possible outcomes  (number of points)  are thus  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.

Consider the two events

  • $A :=$  »The outcome is odd« $  = \{1, 3, 5\}$   ⇒   ${\rm Pr} (A)= 3/6$, 
  • $B :=$  »The outcome is a prime number» $  = \{1, 2, 3, 5\}$   ⇒   ${\rm Pr} (B)= 4/6$.


It can be seen that  $A$  is a  $($proper$)$ subset  of  $B$.  Accordingly,  ${\rm Pr} (A) < {\rm Pr} (B)$  is also true.

Theorems of de Morgan


In many set-theoretical tasks,  the two theorems of  $\text{de Morgan}$  are extremely useful. 

$\text{Theorem of de Morgan:}$

About de Morgan's theorems
$$\overline{A \cup B} = \overline{A} \cap \overline{B},$$
$$\overline{A \cap B} = \overline{A} \cup \overline{B}.$$

These regularities are illustrated in the Venn diagram:

  1. Set  $A$  is shown in red and set  $B$  is shown in blue.
  2. The complement  $\overline {A}$  of  $A$  is hatched in the horizontal direction.
  3. The complement  $\overline {B}$  of  $B$  is hatched in the vertical direction.
  4. The complement  $\overline{A \cup B}$  of the union  ${A \cup B}$  is hatched both horizontally and vertically.
  5. It is thus equal to the intersection  $\overline{A} \cap \overline{B}$  of the two complement sets of  $A$  and  $B$:
$$\overline{A \cup B} = \overline{A} \cap \overline{B}.$$


The second form of the de Morgan theorem can also be illustrated graphically with the same Venn diagram:

  1. The intersection  $A ∩ B$  $($shown in purple in the figure$)$  is neither horizontally nor vertically hatched.
  2. Accordingly, the complement  $\overline{A ∩ B}$  of the intersection is hatched either horizontally, vertically, or in both directions.
  3. By de Morgan's second theorem,  the complement of the intersection equals the union of the two complementary sets of  $A$  and  $B$:
$$\overline{A \cap B} = \overline{A} \cup \overline{B}.$$

$\text{Example 5:}$  We consider again the experiment  »throwing a die«.  The possible outcomes  (number of points)  are thus  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.

We consider the two sets

  • $A : =$  »The outcome is odd«  $= \{1, 3, 5\}$,
  • $B : =$  »The outcome is greater than  $2$«  $= \{3, 4, 5, 6\}$.


From this follow the two complementary sets

  • $\overline {A} : =$  »The outcome is even«  $= \{2, 4, 6\}$,
  • $\overline {B} : =$  »The outcome is smaller than  $3$«  $= \{1, 2\}$.


Further,  using the above theorems,  we obtain the following sets:

$$\overline{A \cup B} = \overline{A} \cap \overline{B} = \{2\},$$
$$\overline{\it A \cap \it B} =\overline{\it A} \cup \overline{\it B} = \{1,2,4,6\}.$$

Disjoint sets


Disjoint sets in the Venn diagram

$\text{Definition:}$  Two sets  $A$  and  $B$  are called  »disjoint« or  »incompatible«,

  • if there is no single element,
  • that is contained in both  $A$  and  $B$.


The diagram shows two disjoint sets  $A$  and  $B$  in the Venn diagram.

In this special case,  the following statements hold:

  • The intersection of two disjoint sets  $A$  and  $B$  always yields the empty set:
$${\rm Pr}(A \cap B) = {\rm Pr}(\phi) = \rm 0.$$
  • The probability of the union set of two disjoint sets  $A$  and  $B$  is always equal to the sum of the two individual probabilities:
$${\rm Pr}( A \cup B) = {\rm Pr}( A) + {\rm Pr}(B).$$


$\text{Example 6:}$  We consider again the experiment  »throwing a die«.  The possible outcomes  (number of points)  are thus  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.

In our standard experiment,  the two sets are now

  • $A :=$  »The outcome is smaller than  $3$ « $ = \{1, 2\}$   ⇒   ${\rm Pr}( A) = 2/6$,
  • $B :=$  »The outcome is greater than  $3$ « $ = \{4, 5,6\}$   ⇒   ${\rm Pr}( B) = 3/6$


disjoint to each other,  since  $A$  and  $B$  do not contain a single common element.

  1. The intersection yields the empty set:  ${A \cap B} = \phi$.
  2. The probability of the union set  ${A \cup B} = \{1, 2, 4, 5, 6\}$  is equal to  ${\rm Pr}( A) + {\rm Pr}(B) = 5/6.$

Addition rule


Only for disjoint sets  $A$  and  $B$,  the relation  ${\rm Pr}( A \cup B) = {\rm Pr}( A) + {\rm Pr}(B)$  holds for the probability of the union set.  But how is this probability calculated for general events that are not necessarily disjoint?

»Addition rule«  of probability calculus

Consider the right-hand Venn diagram with the intersection  $A ∩ B$  shown in purple:

  1. The red set contains all the elements that belong to  $A$,  but not to  $B$.
  2. The elements of  $B$, that are not simultaneously contained in  $A$  are shown in blue.
  3. All red,  blue,  and purple surfaces together make up the union set  $A ∪ B$.


From this set-theoretical representation,  one can see the following relationships:

$${\rm Pr}(A) \hspace{0.8cm}= {\rm Pr}(A \cap B) + {\rm Pr}(A \cap \overline{B}),$$
$${\rm Pr}(B) \hspace{0.8cm}= {\rm Pr}(A \cap B) \rm +{\rm Pr}(\overline{A} \cap {B}),$$
$${\rm Pr}(A \cup B) ={\rm Pr}(A \cap B) +{\rm Pr} ({A} \cap \overline{B}) + {\rm Pr}(\overline{A} \cap {B}).$$

Adding the first two equations and subtracting from them the third,  we get:

$${\rm Pr}(A) +{\rm Pr}(B) -{\rm Pr}(A \cup B) = {\rm Pr}(A \cap B).$$

$\text{Definition:}$  By rearranging this equation,  one arrives at the so-called  »addition rule«  for any two,  not necessarily disjoint events:

$${\rm Pr}(A \cup B) = {\rm Pr}(A) + {\rm Pr}(B) - {\rm Pr}(A \cap B).$$


$\text{Example 7:}$  We consider again the experiment  »throwing a die«.  The possible outcomes  (number of points)  are thus  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.

We consider the two sets

  • $A :=$  »The outcome is odd « $= \{1, 3, 5\}$   ⇒   ${\rm Pr}(A) = 3/6$,
  • $B :=$  »The outcome is greater than  $2$ « $ = \{3, 4, 5, 6\}$   ⇒   ${\rm Pr}(B) = 4/6$.


This gives the following probabilities

  • of the union   ⇒   ${\rm Pr}(A ∪ B) = 5/6$,  and
  • of the intersection   ⇒   ${\rm Pr}(A ∩ B) = 2/6$.


The numerical values show the validity of the addition rule:  

$$5/6 = 3/6 + 4/6 − 2/6.$$

Complete system


In the last section to this chapter,  we consider again more than two possible events, namely, in general,  $I$.  These events will be denoted by  $A_i$   ⇒   the running index $i$ can be in the range  $1 ≤ i ≤ I$.

$\text{Definition:}$  A constellation with events  $A_1, \hspace{0.1cm}\text{...}\hspace{0.1cm} , A_i, \hspace{0.1cm}\text{...}\hspace{0.1cm} , A_I$  is called a  »complete system«,  if and only if the following two conditions are satisfied:

(1)   All events are pairwise disjoint:

$$A_i \cap A_j = \it \phi \hspace{0.25cm}\rm for\hspace{0.15cm}all\hspace{0.25cm}\it i \ne j.$$

(2)   The union of all event sets gives the universal set:

$$\bigcup_{i=1}^{I} A_i = G.$$


Given these two assumptions, the sum of all probabilities is then:

$$\sum_{i =1}^{ I} {\rm Pr}(A_i) = 1.$$

$\text{Example 8:}$ 

  • The sets  $A_1 := \{1, 5\}$  and  $A_2 := \{2, 3\}$  together with the set  $A_3 := \{4, 6\}$  result in a complete system in the random experiment  »throwing a die«,
  • but not in the experiment  »throwing a roulette ball«.


$\text{Example 9:}$  Another example of a complete system is the discrete random variable  $X = \{ x_1, x_2, \hspace{0.1cm}\text{...}\hspace{0.1cm} , x_I\}$  with the likelihood corresponding to the following  »probability mass function«  $\rm (PMF)$:

$$P_X(X) = \big [ \hspace{0.1cm} P_X(x_1),\ P_X(x_2), \hspace{0.05cm}\text{...}\hspace{0.15cm},\ P_X(x_I) \hspace{0.05cm} \big ] = \big [ \hspace{0.1cm} p_1, p_2, \hspace{0.05cm}\text{...} \hspace{0.15cm}, p_I \hspace{0.05cm} \big ] \hspace{0.05cm}$$
$$\Rightarrow \hspace{0.3cm} p_1 = P_X(x_1) = {\rm Pr}(X=x_1) \hspace{0.05cm}, \hspace{0.2cm}p_2 = {\rm Pr}(X=x_2) \hspace{0.05cm},\hspace{0.05cm}\text{...}\hspace{0.15cm},\hspace{0.2cm} p_I = {\rm Pr}(X=x_I) \hspace{0.05cm}.$$
  • The possible outcomes  $x_i$  of the random variable  $X$  are pairwise disjoint to each other.
  • The sum of all likelihoods  $p_1 + p_2 + \hspace{0.1cm}\text{...}\hspace{0.1cm} + \hspace{0.05cm} p_I$  always yields the result  $1$.


$\text{Example 10:}$  Let  $X= \{0, 1, 2 \}$  and  $P_X (X) = \big[0.2, \ 0.5, \ 0.3\big]$. Then holds:

$${\rm Pr}(X=0) = 0.2 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X=1) = 0.5 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X=2) = 0.3 \hspace{0.05cm}.$$

With random variable  $X = \{1, \pi, {\rm e} \}$  and the same  $P_X(X) = \big[0.2, \ 0.5, \ 0.3\big]$  the assignments are:

$${\rm Pr}(X=1) = 0.2 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X=\pi) = 0.5 \hspace{0.05cm}, \hspace{0.2cm} {\rm Pr}(X={\rm e}) = 0.3 \hspace{0.05cm}.$$

$\text{Hints:}$

  • The  »probability mass function«  $P_X(X)$  only makes statements about probabilities,  not about the set of values  $\{x_1, x_2, \hspace{0.1cm}\text{...}\hspace{0.1cm} , x_I\}$  of the random variable  $X$.

Exercises for the chapter


Exercise 1.2: Switching Logic (D/B Converter)

Exercise 1.2Z: Sets of Digits

Exercise 1.3: Fictional University Somewhere

Exercise 1.3Z: Winning with Roulette?