book:chap2:2.2_sets
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book:chap2:2.2_sets [2021/10/26 11:33] – abril | 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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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> | + | <wrap lo> |
- | 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 $\land$, $\lor$, $\Rightarrow$, | ||
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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 | ||
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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 [[# | + | |
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: |
book/chap2/2.2_sets.1635240837.txt.gz · Last modified: 2021/10/26 11:33 by abril