Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

User Tools

Site Tools

Trace:
book:chap2:2.2_sets

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revisionPrevious revision
Next revision		Previous revision
book:chap2:2.2_sets [2021/10/25 23:20] – jvbook:chap2:2.2_sets [2022/04/01 20:00] (current) – jv
Line 1: Line 1:
 +[[forcestorques|2. Balancing Forces and Torques]]
 +  * [[  2.1 Motivation and Outline| 2.1 Motivation and outline: forces are vectors ]]
 +  * ** 2.2 Sets **
 +  * [[  2.3 Groups| 2.3 Groups ]]
 +  * [[  2.4 Fields| 2.4 Fields ]]
 +  * [[  2.5 Vector spaces| 2.5 Vector spaces ]]
 +  * [[  2.6 Physics application balancing forces| 2.6.  Physics application: balancing forces]]
 +  * [[  2.7 The inner product | 2.7 The inner product]]
 +  * [[  2.8 Cartesian coordinates | 2.8 Cartesian coordinates]]
 +  * [[  2.9 Cross products --- torques| 2.9 Cross products — torques ]]
 +  * [[ 2.10 Worked example Calder's mobiles| 2.10 Worked example: Calder's mobiles ]]
 +  * [[ 2.11 Problems| 2.11 Problems ]]
 +  * [[ 2.12 Further reading| 2.12 Further reading ]]
 +
 +----
 +
 ===== 2.2 Sets ===== ===== 2.2 Sets =====
  
Line 22: Line 38:
  
  
-<WRAP box round>**Example 2.2** <wrap em>Sets</wrap> +<WRAP box round #bsp_sets >**Example 2.2** <wrap em>Sets</wrap> 
 \\  \\ 
   *  Set of capitals of German states: \\   *  Set of capitals of German states: \\
Line 32: Line 48:
  
 The cardinalities of these sets are \\ The cardinalities of these sets are \\
-\centering $|A_C| = 16$, $|A_L = 30|$, and $|A_M| = 12$.+<wrap center> $|A_C| = 16$, $|A_L = 30|$, and $|A_M| = 12$. </wrap>
 </WRAP> </WRAP>
  
-<WRAP box round>**Example 2.3** <wrap em>Sets of sets</wrap> \\ +<WRAP box round #bsp_setsofsets >**Example 2.3** <wrap em>Sets of sets</wrap> \\ 
 A set can be an element of a set. A set can be an element of a set.
 For instance the set For instance the set
Line 59: Line 75:
 In obvious cases we use ellipses such as In obvious cases we use ellipses such as
 $A_L = \{$a, b, c, \dots, z, ä, ö, ü, ß$\}$ $A_L = \{$a, b, c, \dots, z, ä, ö, ü, ß$\}$
-for the set given in \Example{Sets}.+for the set given in  [[#bsp_Sets|Example 2.2]].
 Alternatively, one can provide a set $M$ by specifying the properties $A(x)$ Alternatively, one can provide a set $M$ by specifying the properties $A(x)$
 of its elements $x$ in the following form of its elements $x$ in the following form
Line 87: Line 103:
 implies $\Rightarrow$, and implies $\Rightarrow$, and
 is equivalent $\Leftrightarrow$ is equivalent $\Leftrightarrow$
-for the relations indicated in [[#tab_Junktors|Table .+for the relations indicated in [[#tab_Junktors|Table 2.1]].
  
 <WRAP 400px center #tab_Junktors > <WRAP 400px center #tab_Junktors >
Line 105: Line 121:
 In the rightmost column we state the contents of the expression in the left column in words. In the rightmost column we state the contents of the expression in the left column in words.
 The final three lines provide examples of more complicated expressions. The final three lines provide examples of more complicated expressions.
-<wrap hide>\label{table:Junktors}</wrap> 
 </WRAP> </WRAP>
  
Line 126: Line 141:
 <WRAP box round>**Example 2.5** <wrap em>Subsets</wrap> \\  <WRAP box round>**Example 2.5** <wrap em>Subsets</wrap> \\ 
  
-  *  The set of month with names that end with 'ber' is a subset of the set $A_M$ of \Example{Sets}+  *  The set of month with names that end with 'ber' is a subset of the set $A_M$ of [[#bsp_Sets|Example 2.2]]:
 \begin{align*} \begin{align*}
       \{ \text{September, October, November, December} \} \subseteq A_M       \{ \text{September, October, November, December} \} \subseteq A_M
 \end{align*} \end{align*}
  
-  *  For the set  $M$  of  \Example{SetsOfSets}  one has+  *  For the set  $M$  of  [[#bsp_SetsOfSets|Example 2.3]]  one has
 \begin{align*} \begin{align*}
       \{ 1 \} \subseteq M \, , \quad        \{ 1 \} \subseteq M \, , \quad 
Line 146: Line 161:
  
 Two sets are the same when they are subsets of each other. Two sets are the same when they are subsets of each other.
-<WRAP box round>**Theorem 2.1 <wrap hi>Äquivalenz von Mengen</wrap>** \\ +<WRAP box round #Thm_SetEquivalence >**Theorem 2.1 <wrap em> Equivalence of Sets</wrap>** \\ 
 Two sets  $A$ and $B$   are //equal// or //equivalent//, iff Two sets  $A$ and $B$   are //equal// or //equivalent//, iff
 \begin{align*} \begin{align*}
Line 153: Line 168:
 </WRAP> </WRAP>
  
-<wrap lo>**Anmerkung.** [iff] +<wrap lo>**Remark.** In mathematics "iff" indicates that something holds "if and only if".
-In mathematics ``iff'' indicates that something holds +
-``if and only if''.+
 Observe its use in the following two statements: Observe its use in the following two statements:
 A number is an even number if it is the product of two even numbers. A number is an even number if it is the product of two even numbers.
Line 161: Line 174:
 </wrap> </wrap>
  
-<wrap lo #rem_LogicalPrecedence >**Anmerkung.** [precedence of operations in logical expressions.] +<wrap lo #rem_LogicalPrecedence >**Remark.** In logical expressions we first evaluate $\in$, $\not\in$ and other set operations that are used to build logical expressions. 
-In logical expressions we first evaluate $\in$, $\not\in$ and other set operations +
-that are used to build logical expressions. +
 Then we evaluate the junctor $\lnot$ that is acting on a a single logical expression. Then we evaluate the junctor $\lnot$ that is acting on a a single logical expression.
 Finally the other junctors $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$ are evaluated. Finally the other junctors $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$ are evaluated.
-Hence, the brackets are not required in \Thm{SetEquivalence}.+Hence, the brackets are not required in [[#Thm_SetEquivalence|Theorem 2.1]].
 </wrap> </wrap>
  
-=== Proof of \Thm{SetEquivalence} ===+=== Proof of Theorem 2.1 ===
 $A \subseteq B$ implies that $a \in A \Rightarrow a \in B$.  $A \subseteq B$ implies that $a \in A \Rightarrow a \in B$. 
 $B \subseteq A$ implies $b \in B \Rightarrow b \in A$.  $B \subseteq A$ implies $b \in B \Rightarrow b \in A$. 
Line 175: Line 186:
 $\rule{3mm}{3mm}$ $\rule{3mm}{3mm}$
  
-The description of sets by properties of its members, \Example{SetOfDigits},+The description of sets by properties of its members, [[#bsp_SetOfDigits|Example 2.4]],
 suggests that one will often be interested in operations on sets. suggests that one will often be interested in operations on sets.
 For instance the odd and even numbers are subsets of the natural numbers. For instance the odd and even numbers are subsets of the natural numbers.
Line 184: Line 195:
  
 <WRAP 200px right> <WRAP 200px right>
-<wrap #fig_setIntersection >+<wrap #fig_setIntersection > 
 {{ book:chap2:mengen_schnitt.png?direct&200 |}} {{ book:chap2:mengen_schnitt.png?direct&200 |}}
  
Line 216: Line 227:
   *  //Union//:  $M_1 \bigcup M_2 = \left\{ m   \; \vert \;  m \in M_1 \lor m \in M_2 \right\}$,   *  //Union//:  $M_1 \bigcup M_2 = \left\{ m   \; \vert \;  m \in M_1 \lor m \in M_2 \right\}$,
   *  //Difference//:  $M_1 \backslash M_2 = \left\{ m   \; \vert \;  m \in M_1 \land m \notin M_2 \right\}$,   *  //Difference//:  $M_1 \backslash M_2 = \left\{ m   \; \vert \;  m \in M_1 \land m \notin M_2 \right\}$,
-  *  The //complement// of a set $M$ in a //universe// $U$ is defined for subsets $M \subseteq U$ +  *  The //complement// of a set $M$ in a //universe// $U$ is defined for subsets $M \subseteq U$ as $M^{C} = \left\{ m \in U   \; \vert \;  m \notin M \right\} = U \backslash M$. 
-as   $M^{C} = \left\{ m \in U   \; \vert \;  m \notin M \right\} = U \backslash M$. +  *  The //Cartesian product of two sets $M_1$ and $M_2$// is defined as the set of ordered pairs $(a, \, b)$ of elements $a \in M_1$ and \\ $b \in M_2$:  $\displaystyle M_1 \times M_2 = \left\{ (a, \, b)   \; \vert \;  a \in M_1, \, b \in M_2 \right\} $.
-  *  The //Cartesian product of two sets $M_1$ and $M_2$// is defined as the set of ordered pairs +
-$(a, \, b)$ of elements $a \in M_1$ and \\ $b \in M_2$:  $\displaystyle M_1 \times M_2 = \left\{ (a, \, b)   \; \vert \;  a \in M_1, \, b \in M_2 \right\} $.+
  
 A graphical illustration of the operations is provided in A graphical illustration of the operations is provided in
-\cref{figure:setIntersection,figure:setUnion,figure:setDifference,figure:setComplement}.+[[#fig_setIntersection|Figure 2.2]], [[#fig_setUnion|Figure 2.3]], [[#fig_setDifference|Figure 2.4]], [[#fig_setComplement|Figure 2.5]].
 </WRAP> </WRAP>
  
Line 230: Line 239:
 The set of non-female participants is  $P \backslash F$. The set of non-female participants is  $P \backslash F$.
 The set of heterosexual couples in the class is a subset of the Cartesian product $F \times M$.  The set of heterosexual couples in the class is a subset of the Cartesian product $F \times M$. 
-Furthermore, the union $F \bigcup M$ is a proper subset of$P$,  +Furthermore, the union $F \bigcup M$ is a proper subset of $P$, when there is a participant who is neither female nor male.
-when there is a participant who is neither female nor male.+
 </WRAP> </WRAP>
  
Line 243: Line 251:
  
 <WRAP box round>**Example 2.7** <wrap em>Logical quantors and properties of set elements</wrap> \\  <WRAP box round>**Example 2.7** <wrap em>Logical quantors and properties of set elements</wrap> \\ 
-Let $|m|$ denote the number of days in a month $a \in A_M$  +Let $|m|$ denote the number of days in a month $a \in A_M$ (Refer to [[#bsp_Sets|Example 2.2]]).
-(\cf\Example{Sets}).+
 Then the following statements are true: Then the following statements are true:
 There is exactly one month that has exactly $28$ days:  There is exactly one month that has exactly $28$ days: 
Line 263: Line 270:
  
 Many sets of numbers that are of interest in physics have infinitely many elements. Many sets of numbers that are of interest in physics have infinitely many elements.
-We construct them in \cref{table:SetsOfNumbers} based on the natural numbers+We construct them in [[#tab_SetsOfNumbers|Table 2.2]] based on the natural numbers
 \begin{align*} \begin{align*}
   \mathbb N   \mathbb N
Line 291: Line 298:
 | negative numbers         | $-\mathbb{N}$  | $\left\{ -n   \; \vert \;  n \in \mathbb{N} \right\}$ | | negative numbers         | $-\mathbb{N}$  | $\left\{ -n   \; \vert \;  n \in \mathbb{N} \right\}$ |
 | even numbers             | $2 \mathbb{N}$ | $\left\{ 2\,n   \; \vert \;  n \in \mathbb{N} \right\}$ | | even numbers             | $2 \mathbb{N}$ | $\left\{ 2\,n   \; \vert \;  n \in \mathbb{N} \right\}$ |
-| odd numbers              | $2 \mathbb{N} - 1$   | $\left\{ 2\,n -1  \; \vert \;  n \in \mathbb{N} \right\} |+| odd numbers              | $2 \mathbb{N} - 1$   | $\left\{ 2\,n -1  \; \vert \;  n \in \mathbb{N}\right\}$ |
 | integer numbers        | $\mathbb{Z}$   | $\left( -\mathbb{N} \right) \bigcup \mathbb{N}_0$ | | integer numbers        | $\mathbb{Z}$   | $\left( -\mathbb{N} \right) \bigcup \mathbb{N}_0$ |
 | rational numbers       | $\mathbb{Q}$   | $\left\{ \frac{p}{q}   \; \vert \;  p \in \mathbb{Z}, \, q \in \mathbb{N} \right\}$ | | rational numbers       | $\mathbb{Q}$   | $\left\{ \frac{p}{q}   \; \vert \;  p \in \mathbb{Z}, \, q \in \mathbb{N} \right\}$ |
Line 344: Line 351:
 </WRAP> </WRAP>
 </WRAP> </WRAP>
 +
  
 ---- ----
 +
 <WRAP #quest_setSelftest-intervals > <WRAP #quest_setSelftest-intervals >
 Problem 2.2:  Problem 2.2: 
Line 353: Line 362:
   -  Provide $[-1, 4] \backslash [1, 2[$ as union of two intervals.   -  Provide $[-1, 4] \backslash [1, 2[$ as union of two intervals.
 </WRAP> </WRAP>
 +
 ---- ----
 +
 <WRAP #quest_setSelftest-numbers > <WRAP #quest_setSelftest-numbers >
 Problem 2.3:  Problem 2.3: 
Line 362: Line 373:
   -  $\lbrace 6 \cdot z \vert z \in \mathbb{Z} \rbrace \subset \lbrace 2 \cdot z \vert z \in \mathbb{Z} \rbrace$.   -  $\lbrace 6 \cdot z \vert z \in \mathbb{Z} \rbrace \subset \lbrace 2 \cdot z \vert z \in \mathbb{Z} \rbrace$.
   -  $\left\lbrace 2 \cdot z \vert z \in \mathbb{Z} \right\rbrace \cap \left\lbrace 3 \cdot z \vert z \in \mathbb{Z} \right\rbrace = \left\lbrace 6 \cdot z \vert z \in \mathbb{Z} \right\rbrace$.   -  $\left\lbrace 2 \cdot z \vert z \in \mathbb{Z} \right\rbrace \cap \left\lbrace 3 \cdot z \vert z \in \mathbb{Z} \right\rbrace = \left\lbrace 6 \cdot z \vert z \in \mathbb{Z} \right\rbrace$.
-  -  Let $T(a)$ be the set of numbers that divide $a$. Then +  -  Let $T(a)$ be the set of numbers that divide $a$. Then  $\quad\displaystyle  \forall a, b \in \mathbb{N}  \; \vert  \; T(a) \cup T(b) = T(a \cdot b) $. 
-\begin{align*} +
-      \forall a, b \in \mathbb{N}  \; \vert \;\quad T(a) \cup T(b) = T(a \cdot b)  +
-\end{align*}+
 Example:  $T(2) = \{1,2\}$, $T(3) = \{1,3\}$, and $T(6) = \{1,2,3,6\}$.  Example:  $T(2) = \{1,2\}$, $T(3) = \{1,3\}$, and $T(6) = \{1,2,3,6\}$. 
 +</WRAP>
  
 ~~DISCUSSION|Questions, Remarks, and Suggestions~~ ~~DISCUSSION|Questions, Remarks, and Suggestions~~
book/chap2/2.2_sets.1635196842.txt.gz · Last modified: 2021/10/25 23:20 by jv