book:chap2:2.2_sets
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book:chap2:2.2_sets [2021/10/25 23:22] – jv | book:chap2:2.2_sets [2022/04/01 20:00] (current) – jv | ||
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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 ===== | ||
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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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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> | ||
</ | </ | ||
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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 # | + | <wrap lo # |
- | 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> | ||
- | <wrap # | + | <wrap # |
{{ book: | {{ book: | ||
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* // | * // | ||
* // | * // | ||
- | * The // | + | * The // |
- | as | + | * 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: | + | [[# |
</ | </ | ||
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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. | + | |
</ | </ | ||
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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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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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| negative numbers | | negative numbers | ||
| even numbers | | even numbers | ||
- | | odd numbers | + | | odd numbers |
| integer numbers | | integer numbers | ||
| rational numbers | | rational numbers | ||
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</ | </ | ||
</ | </ | ||
+ | |||
---- | ---- | ||
+ | |||
<WRAP # | <WRAP # | ||
Problem 2.2: | Problem 2.2: | ||
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- Provide $[-1, 4] \backslash [1, 2[$ as union of two intervals. | - Provide $[-1, 4] \backslash [1, 2[$ as union of two intervals. | ||
</ | </ | ||
+ | |||
---- | ---- | ||
+ | |||
<WRAP # | <WRAP # | ||
Problem 2.3: | Problem 2.3: | ||
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- $\lbrace 6 \cdot z \vert z \in \mathbb{Z} \rbrace \subset \lbrace 2 \cdot z \vert z \in \mathbb{Z} \rbrace$. | - $\lbrace 6 \cdot z \vert z \in \mathbb{Z} \rbrace \subset \lbrace 2 \cdot z \vert z \in \mathbb{Z} \rbrace$. | ||
- $\left\lbrace 2 \cdot z \vert z \in \mathbb{Z} \right\rbrace \cap \left\lbrace 3 \cdot z \vert z \in \mathbb{Z} \right\rbrace = \left\lbrace 6 \cdot z \vert z \in \mathbb{Z} \right\rbrace$. | - $\left\lbrace 2 \cdot z \vert z \in \mathbb{Z} \right\rbrace \cap \left\lbrace 3 \cdot z \vert z \in \mathbb{Z} \right\rbrace = \left\lbrace 6 \cdot z \vert z \in \mathbb{Z} \right\rbrace$. | ||
- | - 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 |
- | \begin{align*} | + | |
- | | + | |
- | \end{align*} | + | |
Example: | Example: | ||
+ | </ | ||
~~DISCUSSION|Questions, | ~~DISCUSSION|Questions, |
book/chap2/2.2_sets.1635196966.txt.gz · Last modified: 2021/10/25 23:22 by jv