Theory of Stochastic Signals/Set Theory Basics - Revision history

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@@ Line 184: / Line 184: @@
 {{BlaueBox|TEXT=
-&raquo;$\text{Theorem of de Morgan:}$&laquo;
+$\text{Theorem of de Morgan:}$
 [[File:EN_Sto_T_1_2_S6.png|frame| Zu den Theoremen von de Morgan | About de Morgan's theorems]]

@@ Line 181: / Line 181: @@
 ==Theorems of de Morgan==
 <br>
 [[File:EN_Sto_T_1_2_S6.png|frame| Zu den Theoremen von de Morgan | About de Morgan's theorems]]
-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; These are:
 :$$\overline{A \cup B} = \overline{A} \cap \overline{B},$$
 :$$\overline{A \cap B} = \overline{A} \cup \overline{B}.$$
 These regularities are illustrated in the Venn diagram:
-*Set&nbsp; $A$&nbsp; is shown in red and set&nbsp; $B$&nbsp; is shown in blue.
+#Set&nbsp; $A$&nbsp; is shown in red and set&nbsp; $B$&nbsp; is shown in blue.
-*The complement&nbsp; $\overline {A}$&nbsp; of&nbsp; $A$&nbsp; is hatched in the horizontal direction.
+#The complement&nbsp; $\overline {A}$&nbsp; of&nbsp; $A$&nbsp; is hatched in the horizontal direction.
-*The complement&nbsp;  $\overline {B}$&nbsp; of&nbsp; $B$&nbsp; is hatched in the vertical direction.
+#The complement&nbsp;  $\overline {B}$&nbsp; of&nbsp; $B$&nbsp; is hatched in the vertical direction.
-*The complement&nbsp; $\overline{A \cup B}$&nbsp; of the union&nbsp; ${A \cup B}$&nbsp; is hatched both horizontally and vertically.
+#The complement&nbsp; $\overline{A \cup B}$&nbsp; of the union&nbsp; ${A \cup B}$&nbsp; is hatched both horizontally and vertically.
-*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; $B$:
+#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; $B$:
-:$$\overline{A \cup B} = \overline{A} \cap \overline{B}.$$
+::$$\overline{A \cup B} = \overline{A} \cap \overline{B}.$$}}
 The second form of the de Morgan theorem can also be illustrated graphically with the same Venn diagram:
-*The intersection&nbsp; $A ∩ B$&nbsp; $($shown in purple in the figure$)$&nbsp; is neither horizontally nor vertically hatched.
+#The intersection&nbsp; $A ∩ B$&nbsp; $($shown in purple in the figure$)$&nbsp; is neither horizontally nor vertically hatched.
-*Accordingly, the complement&nbsp; $\overline{A ∩ B}$&nbsp; of the intersection is hatched either horizontally, vertically, or in both directions.
+#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, the complement of the intersection equals the union of the two complementary sets of&nbsp; $A$&nbsp; and&nbsp; $B$:
+#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; $B$:
-:$$\overline{A \cap B} = \overline{A} \cup \overline{B}.$$
+::$$\overline{A \cap B} = \overline{A} \cup \overline{B}.$$
 {{GraueBox|TEXT=
-$\text{Example 5:}$&nbsp; We consider again the experiment&nbsp; "throwing a die".&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
+$\text{Example 5:}$&nbsp; We consider again the experiment&nbsp; &raquo;throwing a die&laquo;.&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
 We consider the two sets
-* $A : =$&nbsp; "The outcome is odd" $= \{1, 3, 5\}$,
+* $A : =$&nbsp; &raquo;The outcome is odd&laquo;&nbsp; $= \{1, 3, 5\}$,
-* $B : =$&nbsp; "The outcome is greater than&nbsp; $2$" $= \{3, 4, 5, 6\}$.
+* $B : =$&nbsp; &raquo;The outcome is greater than&nbsp; $2$&laquo;&nbsp; $= \{3, 4, 5, 6\}$.
 From this follow the two complementary sets
-* $\overline {A} : =$&nbsp; "The outcome is even" $= \{2, 4, 6\}$,
+* $\overline {A} : =$&nbsp; &raquo;The outcome is even&laquo;&nbsp; $= \{2, 4, 6\}$,
-* $\overline {B} : =$&nbsp; "The outcome is smaller than&nbsp; $3$" $= \{1, 2\}$.
+* $\overline {B} : =$&nbsp; &raquo;The outcome is smaller than&nbsp; $3$&laquo;&nbsp; $= \{1, 2\}$.

@@ Line 135: / Line 135: @@
 ==Proper subset &ndash; Improper subset==
 <br>
-[[File:EN_Sto_T_1_2_S5.png | right|frame| Proper subset in the Venn diagram]]
+{{BlaueBox|TEXT=
-{{BlaueBox|TEXT=
+[[File:EN_Sto_T_1_2_S5.png | right|frame| Proper subset in the Venn diagram]]
 $\text{Definitions:}$&nbsp;
 '''(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 this case,&nbsp; the probabilities are:
+In this case,&nbsp; 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)'''&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
@@ Line 153: / Line 155: @@
 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.
-*The relation&nbsp;  ${\rm Pr} (A) \le  {\rm Pr} (B)$&nbsp; then applies to the probabilities.
+*Then applies to the probabilities:&nbsp;  ${\rm Pr} (A) \le  {\rm Pr} (B)$.
 *The equality sign is only valid for the special case&nbsp;  $A = B$.}}
-In addition,&nbsp; however,&nbsp; the two equations known as the&nbsp; &raquo;'''absorption laws'''&laquo;&nbsp; also apply:
+In addition,&nbsp; the two equations known as the&nbsp; &raquo;'''absorption laws'''&laquo;&nbsp; also apply:
 :$$(A \cap B)  \cup A  =  A ,$$
 :$$(A  \cup B) \cap A  =  A,$$
-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; $A ∪ B$.
+*since the intersection&nbsp; $A ∩ B$&nbsp; is always a subset of&nbsp; $A$,&nbsp;
 {{GraueBox|TEXT=
-$\text{Example 4:}$&nbsp; We consider again the experiment&nbsp; "throwing a die".&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
+$\text{Example 4:}$&nbsp; We consider again the experiment&nbsp; &raquo;throwing a die&laquo;.&nbsp; The possible outcomes&nbsp; (number of points)&nbsp; are thus&nbsp;  $E_μ ∈ G = \{1, 2, 3, 4, 5, 6\}$.
 Consider the two events
-* $A :=$&nbsp; "The outcome is odd" $ = \{1, 3, 5\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A)= 3/6$,&nbsp;  and
+* $A :=$&nbsp; &raquo;The outcome is odd&laquo; $&nbsp; = \{1, 3, 5\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (A)= 3/6$,&nbsp;
-* $B :=$&nbsp; "The outcome is a prime number" $= \{1, 2, 3, 5\}$ &nbsp; &rArr; &nbsp; ${\rm Pr} (B)= 4/6$.
-It can be seen that&nbsp;  $A$&nbsp; is a&nbsp; (proper) subset&nbsp; of&nbsp; $B$&nbsp;.&nbsp; Accordingly,&nbsp; ${\rm Pr} (A) <  {\rm Pr} (B)$&nbsp; is also true. }}
+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==

@@ Line 7: / Line 7: @@
 ==Venn diagram, universal and empty set==
 <br>
 [[File:EN_Sto_T_1_2_S1.png | right|frame|Set representation in the Venn diagram]]
 An important tool of set theory is the&nbsp; &raquo;'''Venn diagram'''&laquo;&nbsp; according to the graph:
-*Applied to probability theory,&nbsp; here the events&nbsp; $A_i$&nbsp; are represented 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;
+*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;
-Note that not every event&nbsp; $A$&nbsp; with&nbsp; ${\rm Pr}(A) = 0$&nbsp; can really never happen.&nbsp; For example:
+*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.
-*The event&nbsp; "The noise value&nbsp; $n$&nbsp; is identically zero"&nbsp; is vanishingly small and&nbsp; ${\rm Pr}(n \equiv 0) = 0$,&nbsp; if&nbsp; $n$&nbsp; is described by a continuous&nbsp; (Gaussian)&nbsp; random variable.
-*Nevertheless,&nbsp; it is of course possible&nbsp; (although extremely unlikely)&nbsp; that at some point the exact noise value&nbsp; $n = 0$&nbsp; will also occur.
 ==Union set==

@@ Line 315: / Line 315: @@
 Hints:
-*The&nbsp; [[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Probability_mass_function_and_probability_density_function|$\text{probability mass function}$]]&nbsp; $P_X(X)$&nbsp; only makes statements about the 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; $X$.
+*The&nbsp; [[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Probability_mass_function_and_probability_density_function|$\text{probability mass function}$]]&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|$\text{probability density function}$]]&nbsp; $\rm (PDF)$.}}

@@ Line 316: / Line 316: @@
 Hints:
 *The&nbsp; [[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Probability_mass_function_and_probability_density_function|$\text{probability mass function}$]]&nbsp; $P_X(X)$&nbsp; only makes statements about the 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_der_Wahrscheinlichkeitsdichtefunktion|$\text{probability density function}$]]&nbsp; $\rm (PDF)$.}}
+*This additional information is provided by the&nbsp; [[Theory_of_Stochastic_Signals/Probability_Density_Function_(PDF)#Definition_of_the_probability_density_function|$\text{probability density function}$]]&nbsp; $\rm (PDF)$.}}
 ==Exercises for the chapter==

@@ Line 238: / Line 238: @@
 But how is this probability calculated for general events that are not necessarily disjoint?
-[[File:EN_Sto_T_1_2_S8.png | right|frame| About the&nbsp; "Addition Theorem"&nbsp; of probability calculus]]
+[[File:EN_Sto_T_1_2_S8.png | right|frame| About the&nbsp; "addition rule"&nbsp; of probability calculus]]
 Consider the right-hand Venn diagram with the intersection&nbsp; $A ∩ B$&nbsp; shown in purple.
 *The red set contains all the elements that belong to&nbsp; $A$,&nbsp; but not to&nbsp; $B$.
@@ Line 254: / Line 254: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; By rearranging this equation, one arrives at the so-called&nbsp; &raquo;'''Addition Rule'''&laquo;&nbsp; for any two, not necessarily disjoint events:
+$\text{Definition:}$&nbsp; By rearranging this equation, one arrives at the so-called&nbsp; &raquo;'''addition rule'''&laquo;&nbsp; for any two, not necessarily disjoint events:
 :$${\rm Pr}(A \cup B) = {\rm Pr}(A) + {\rm Pr}(B) - {\rm Pr}(A \cap B).$$}}
@@ Line 271: / Line 271: @@
-The numerical values show the validity of the addition theorem: &nbsp; $5/6 = 3/6 + 4/6 − 2/6$.}}
+The numerical values show the validity of the addition rule: &nbsp; $5/6 = 3/6 + 4/6 − 2/6$.}}
 ==Complete system==