book:chap2:2.2_sets
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+ | [[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: | ||
+ | * [[ 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' | ||
+ | * [[ 2.11 Problems| 2.11 Problems ]] | ||
+ | * [[ 2.12 Further reading| 2.12 Further reading ]] | ||
+ | |||
+ | ---- | ||
+ | |||
===== 2.2 Sets ===== | ===== 2.2 Sets ===== | ||
In mathematics and physics we often wish to make statements about a collection of objects, numbers, or other distinct entities. | In mathematics and physics we often wish to make statements about a collection of objects, numbers, or other distinct entities. | ||
- | <WRAP box round> | + | <WRAP box round> |
A //set// is a gathering of well-defined, | A //set// is a gathering of well-defined, | ||
\\ | \\ | ||
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<WRAP lo> | <WRAP lo> | ||
- | * When a set $M$ has a finite number of elements, e.g., $+1$ and $-1$, one can specify the elements by explicitly stating the elements, $M = \{ +1, -1 \}$. In which order they are states does not play a role, | + | * When a set $M$ has a finite number of elements, e.g., $+1$ and $-1$, one can specify the elements by explicitly stating the elements, $M = \{ +1, -1 \}$. In which order they are states does not play a role, and it also does not make a difference when elements are provided several times. In other words the set $M$ of cardinality two can be specified by any of the following statements |
- | and it also does not make a difference when elements are provided several times. In other words the set $M$ of cardinality two can be specified by any of the following statements | + | |
\[ | \[ | ||
M = \{ -1, +1 \} = \{ +1, -1 \} = \{ -1, 1, 1, 1, \} = \{ -1, 1, +1, -1 \} | M = \{ -1, +1 \} = \{ +1, -1 \} = \{ -1, 1, 1, 1, \} = \{ -1, 1, +1, -1 \} | ||
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- | <WRAP box round> | + | <WRAP box round # |
\\ | \\ | ||
* Set of capitals of German states: \\ | * Set of capitals of German states: \\ | ||
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The cardinalities of these sets are \\ | The cardinalities of these sets are \\ | ||
- | \centering | + | <wrap center> |
</ | </ | ||
- | <WRAP box round> | + | <WRAP box round # |
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 | ||
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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 |
Alternatively, | Alternatively, | ||
of its elements $x$ in the following form | of its elements $x$ in the following form | ||
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The properties are separated by commas, and must all be true for all elements of the set. | The properties are separated by commas, and must all be true for all elements of the set. | ||
- | <WRAP box round> | + | <WRAP box round # |
The set of digits $D = \left\{ 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9 \right\}$ | The set of digits $D = \left\{ 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9 \right\}$ | ||
can also be defined as follows | can also be defined as follows | ||
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implies $\Rightarrow$, | implies $\Rightarrow$, | ||
is equivalent $\Leftrightarrow$ | is equivalent $\Leftrightarrow$ | ||
- | for the relations indicated in \ref{table: | + | for the relations indicated in [[# |
- | <WRAP 400px center> | + | <WRAP 400px center |
^ $A$ ^ $0$ ^ $0$ ^ $1$ ^ $1$ ^ ^ | ^ $A$ ^ $0$ ^ $0$ ^ $1$ ^ $1$ ^ ^ | ||
^ $B$ ^ $0$ ^ $1$ ^ $0$ ^ $1$ ^ ^ | ^ $B$ ^ $0$ ^ $1$ ^ $0$ ^ $1$ ^ ^ | ||
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|$ \lnot A \land B | |$ \lnot A \land B | ||
|$ A \land \lnot B | |$ A \land \lnot B | ||
- | List of the results of different junctors acting on two statements $A$ and $B$. | + | Table 2.1: List of the results of different junctors acting on two statements $A$ and $B$. |
Here $0$ and $1$ indicate that a statement is wrong or right, respectively. | Here $0$ and $1$ indicate that a statement is wrong or right, respectively. | ||
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> | ||
</ | </ | ||
- | The definition of the digits in \Example{SetOfDigits} | + | The definition of the digits in [[# |
we say that $D$ is a subset of $\mathbb{Z}$. | we say that $D$ is a subset of $\mathbb{Z}$. | ||
- | <WRAP box round> | + | <WRAP box round> |
The set $M_1$ is a //subset// of $M_2, | The set $M_1$ is a //subset// of $M_2, | ||
if all elements of $M_1$ are also contained in $M_2$. | if all elements of $M_1$ are also contained in $M_2$. | ||
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</ | </ | ||
- | <WRAP box round> | + | <WRAP box round> |
- | * The set of month with names that end with ' | + | * The set of month with names that end with ' |
\begin{align*} | \begin{align*} | ||
\{ \text{September, | \{ \text{September, | ||
\end{align*} | \end{align*} | ||
- | * For the set $M$ of | + | * For the set $M$ of |
\begin{align*} | \begin{align*} | ||
\{ 1 \} \subseteq M \, , \quad | \{ 1 \} \subseteq M \, , \quad | ||
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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> | + | <WRAP box round # |
Two sets $A$ and $B$ are //equal// or // | Two sets $A$ and $B$ are //equal// or // | ||
\begin{align*} | \begin{align*} | ||
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</ | </ | ||
- | <wrap lo>**Anmerkung.** [iff] | + | <wrap lo>**Remark.** In mathematics |
- | In mathematics | + | |
- | ``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. | ||
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</ | </ | ||
- | <wrap lo>**Anmerkung.** [precedence of operations in logical expressions.] | + | <wrap lo # |
- | \label{remark: | + | |
- | 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 $\land$, $\lor$, $\Rightarrow$, | Finally the other junctors $\land$, $\lor$, $\Rightarrow$, | ||
- | Hence, the brackets are not required in \Thm{SetEquivalence}. | + | Hence, the brackets are not required in [[# |
</ | </ | ||
- | === 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 $b \in B \Rightarrow b \in A$. | ||
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$\rule{3mm}{3mm}$ | $\rule{3mm}{3mm}$ | ||
- | The description of sets by properties of its members, | + | The description of sets by properties of its members, |
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. | ||
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<WRAP 200px right> | <WRAP 200px right> | ||
- | {{ en:book:chap02: | + | <wrap # |
+ | {{ book:chap2: | ||
- | Intersection of two sets. \\ $\quad$ | + | Figure 2.2: Intersection of two sets. \\ $\quad$ |
- | <wrap hide> | + | |
- | \label{figure: | + | |
</ | </ | ||
- | {{ en:book:chap02: | + | <wrap # |
+ | {{ book:chap2: | ||
- | Union of two sets. \\ $\quad$ | + | Figure 2.3: Union of two sets. \\ $\quad$ |
- | <wrap hide> | + | |
- | \label{figure: | + | |
</ | </ | ||
- | {{ en:book:chap02: | + | <wrap # |
+ | {{ book:chap2: | ||
- | Difference of two sets. \\ $\quad$ | + | Figure 2.4: Difference of two sets. \\ $\quad$ |
- | <wrap hide> | + | |
- | \label{figure: | + | |
</ | </ | ||
- | {{ en:book:chap02: | + | <wrap # |
+ | {{ book:chap2: | ||
- | Complement of a set. \\ $\quad$ | + | Figure 2.5: Complement of a set. \\ $\quad$ |
- | <wrap hide> | + | |
- | \label{figure: | + | |
</ | </ | ||
</ | </ | ||
- | <WRAP box round> | + | <WRAP box round # |
For two sets $M_1$ and $M_2$ we define the following operations: | For two sets $M_1$ and $M_2$ we define the following operations: | ||
- | * // | + | * // |
- | + | * // | |
- | $M_1 \bigcap M_2 = \left\{ m \; \vert \; m \in M_1 \land m \in M_2 \right\}$, | + | * // |
- | * //Union//: | + | * The // |
- | + | * 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\} $. | |
- | $M_1 \bigcup M_2 = \left\{ m \; \vert \; m \in M_1 \lor m \in M_2 \right\}$, | + | |
- | * // | + | |
- | + | ||
- | $M_1 \backslash M_2 = \left\{ m \; \vert \; m \in M_1 \land m \notin M_2 \right\}$, | + | |
- | * The // | + | |
- | 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\} | + | |
- | $. | + | |
A graphical illustration of the operations is provided in | A graphical illustration of the operations is provided in | ||
- | \cref{figure: | + | [[# |
</ | </ | ||
- | <WRAP box round> | + | <WRAP box round> |
Consider the set of participants $P$ in my class. | Consider the set of participants $P$ in my class. | ||
The sets of female $F$ and male $M$ participants of the class are proper subsets of $P$ with an empty intersection $F \bigcap M$. | The sets of female $F$ and male $M$ participants of the class are proper subsets of $P$ with an empty intersection $F \bigcap M$. | ||
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, | + | Furthermore, |
- | when there is a participant who is neither female nor male. | + | |
</ | </ | ||
- | <WRAP box round> | + | <WRAP box round> |
A logical statements $\mathsf S$ about elements $a$ of a set $A$ may hold | A logical statements $\mathsf S$ about elements $a$ of a set $A$ may hold | ||
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</ | </ | ||
- | <WRAP box round> | + | <WRAP box round> |
- | 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 [[# |
- | (\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: | ||
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</ | </ | ||
- | ==== 1.2.1 Sets of Numbers ==== | + | ==== 2.2.1 Sets of Numbers ==== |
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: | + | We construct them in [[# |
\begin{align*} | \begin{align*} | ||
\mathbb N | \mathbb N | ||
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are handy when it comes to problems involving three-dimensional rotations. | are handy when it comes to problems involving three-dimensional rotations. | ||
In any case one needs intervals of numbers. | In any case one needs intervals of numbers. | ||
- | <WRAP center> | + | <WRAP center |
^ name ^ symbol | ^ name ^ symbol | ||
| natural numbers | $\mathbb{N}$ | | natural numbers | $\mathbb{N}$ | ||
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| negative numbers | | negative numbers | ||
| even numbers | | even numbers | ||
- | | odd numbers | + | | odd numbers |
| integer numbers | | integer numbers | ||
| rational numbers | | rational numbers | ||
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| complex numbers | | complex numbers | ||
- | Summary of important sets of numbers. | + | Table 2.2: Summary of important sets of numbers. |
- | </ | + | |
- | \label{table: | + | |
- | </ | + | |
</ | </ | ||
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- | ==== 1.2.2 Self Test ==== | + | ==== 2.2.2 Self Test ==== |
- | Problem | + | <WRAP # |
+ | Problem | ||
** Relations between sets ** | ** Relations between sets ** | ||
- | \\ | + | |
Let $A$, $B$, $C$, and $D$ be pairwise distinct elements. | Let $A$, $B$, $C$, and $D$ be pairwise distinct elements. | ||
Select one of the symbols | Select one of the symbols | ||
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</ | </ | ||
</ | </ | ||
+ | </ | ||
+ | |||
---- | ---- | ||
- | Problem | + | <WRAP # |
+ | Problem | ||
** Intervals ** | ** Intervals ** | ||
- | \\ | ||
- Provide $\lbrack 1; 17 \rbrack \cap \rbrack 0; 5 \lbrack$ as a single interval. | - Provide $\lbrack 1; 17 \rbrack \cap \rbrack 0; 5 \lbrack$ as a single interval. | ||
- Provide $[-1, 4] \backslash [1, 2[$ as union of two intervals. | - Provide $[-1, 4] \backslash [1, 2[$ as union of two intervals. | ||
+ | </ | ||
+ | |||
+ | ---- | ||
- | \bigskip | + | <WRAP # |
- | Problem | + | Problem |
** Sets of numbers ** | ** Sets of numbers ** | ||
- | \\ | + | |
Which of the following statements are true? | Which of the following statements are true? | ||
- $\lbrace 6 \cdot z \vert z \in \mathbb{Z} \rbrace \subset \lbrace 2 \cdot z \vert z \in \mathbb{Z} \rbrace$. | - $\lbrace 6 \cdot z \vert z \in \mathbb{Z} \rbrace \subset \lbrace 2 \cdot z \vert z \in \mathbb{Z} \rbrace$. | ||
- | - $\left\lbrace 2 \cdot z \vert z \in \mathbb{Z} \right\rbrace | + | - $\left\lbrace 2 \cdot z \vert z \in \mathbb{Z} \right\rbrace \cap \left\lbrace 3 \cdot z \vert z \in \mathbb{Z} \right\rbrace = \left\lbrace 6 \cdot z \vert z \in \mathbb{Z} \right\rbrace$. |
- | \cap \left\lbrace 3 \cdot z \vert z \in \mathbb{Z} \right\rbrace | + | - Let $T(a)$ be the set of numbers that divide $a$. Then $\quad\displaystyle |
- | = \left\lbrace 6 \cdot z \vert z \in \mathbb{Z} \right\rbrace$. | + | |
- | - Let $T(a)$ be the set of numbers that divide $a$. Then | + | Example: |
- | \begin{align*} | + | </ |
- | | + | |
- | \end{align*} | + | |
- | {Example: | + | |
~~DISCUSSION|Questions, | ~~DISCUSSION|Questions, |
book/chap2/2.2_sets.1633592310.txt.gz · Last modified: 2021/10/07 09:38 by jv