Theory of Stochastic Signals/Set Theory Basics

1. Theory of Stochastic Signals
2. Probability Calculation
3. Set Theory Basics

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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.

• The upper bound holds for  »disjoint sets«.

$\text{Example 1:}$  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.

• The lower bound is obtained for the joint probability of  »disjoint sets«.

$\text{Example 2:}$  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,  the probabilities are:

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

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.

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

In addition,  however,  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\}$   ⇒   ${\rm Pr} (A)= 3/6$,  and
• $B :=$  "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

About de Morgan's theorems

In many set-theoretical tasks,  the two theorems of  $\text{de Morgan}$  are extremely useful.  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  $A$  is shown in red and set  $B$  is shown in blue.
• The complement  $\overline {A}$  of  $A$  is hatched in the horizontal direction.
• The complement  $\overline {B}$  of  $B$  is hatched in the vertical direction.
• The complement  $\overline{A \cup B}$  of the union  ${A \cup B}$  is hatched both horizontally and vertically.
• 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:

• The intersection  $A ∩ B$  $($shown in purple in the figure$)$  is neither horizontally nor vertically hatched.
• Accordingly, the complement  $\overline{A ∩ B}$  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  $A$  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

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

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

Disjoint sets in the Venn diagram

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.

• The intersection yields the empty set:  ${A \cap B} = \phi$.
• 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?

About the  "addition rule"  of probability calculus

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

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

From this set-theoretic 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  $\text{probability mass function}$  $\rm (PMF)$:

$$P_X(X) = \big [ \hspace{0.1cm} P_X(x_1),\ P_X(x_2), \hspace{0.05cm}\text{...}\hspace{0.15cm},\ P_X(x_I) \hspace{0.05cm} \big ] = \big [ \hspace{0.1cm} p_1, p_2, \hspace{0.05cm}\text{...} \hspace{0.15cm}, p_I \hspace{0.05cm} \big ] \hspace{0.05cm}$$

$$\Rightarrow \hspace{0.3cm} p_1 = P_X(x_1) = {\rm Pr}(X=x_1) \hspace{0.05cm}, \hspace{0.2cm}p_2 = {\rm Pr}(X=x_2) \hspace{0.05cm},\hspace{0.05cm}\text{...}\hspace{0.15cm},\hspace{0.2cm} p_I = {\rm Pr}(X=x_I) \hspace{0.05cm}.$$

• 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:

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

Hints:

• The  $\text{probability mass function}$  $P_X(X)$  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  $X$.
• This additional information is provided by the  $\text{probability density function}$  $\rm (PDF)$.

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?