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real_coordinate_space [2013/09/03 16:30] nikolaj |
real_coordinate_space [2014/04/04 16:01] nikolaj |
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| ===== Real coordinate space ===== | ===== Real coordinate space ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $ n\in \mathbb N $ | | + | | @#55CCEE: context | @#55CCEE: $ n\in \mathbb N $ | |
| + | | @#FFBB00: definiendum | @#FFBB00: $\mathbb R^n$ | | ||
| - | | @#FFBB00: $\mathbb R^n$ | | + | ==== Discussion ==== |
| + | Explicitly, | ||
| - | Recursive definition: | + | $ \mathbb R^1=\mathbb R $ |
| + | $ \mathbb R^p=\mathbb R^{p-1}\times \mathbb R, \hspace{1cm}$ | ||
| - | | $ p\in \mathbb N $ | $ p>1 $ | | + | | @#DDDDDD: range | @#DDDDDD: $1\le i\le p$ | |
| - | | @#55EE55: $ \mathbb R^1=\mathbb R $ | | + | | @#EEEE55: predicate | @#EEEE55: $ a< b \equiv \forall i.\ a_i< b_i$ | |
| - | | @#55EE55: $ \mathbb R^p=\mathbb R^{p-1}\times \mathbb R, \hspace{1cm}$ | | + | | @#EEEE55: predicate | @#EEEE55: $ a> b \equiv \forall i.\ a_i> b_i$ | |
| + | | @#EEEE55: predicate | @#EEEE55: $ a\le b \equiv \forall i.\ a_i\le b_i$ | | ||
| + | | @#EEEE55: predicate | @#EEEE55: $ a\ge b \equiv \forall i.\ a_i\ge b_i$ | | ||
| - | ==== Discussion ==== | + | >todo: define the following in a seperate entry: |
| - | | @#DDDDDD: $1\le i\le p$ | | + | |
| + | $ [a,b]\equiv\{x\ |\ \forall i.\ a_i\le x_i\le b_i\}$ | ||
| + | |||
| + | $ ]a,b]\equiv\{x\ |\ \forall i.\ a_i<x_i\le b_i\}$ | ||
| - | | @#EEEE55: $ a< b \equiv \forall i.\ a_i< b_i$ | | + | $ [a,b[\ \equiv\{x\ |\ \forall i.\ a_i\le x_i<b_i\}$ |
| - | | @#EEEE55: $ a> b \equiv \forall i.\ a_i> b_i$ | | + | |
| - | | @#EEEE55: $ a\le b \equiv \forall i.\ a_i\le b_i$ | | + | |
| - | | @#EEEE55: $ a\ge b \equiv \forall i.\ a_i\ge b_i$ | | + | |
| - | | @#FFBB00: $ [a,b]\equiv\{x\ |\ \forall i.\ a_i\le x_i\le b_i\}$ | | + | $ ]a,b[\ \equiv\{x\ |\ \forall i.\ a_i<x_i<b_i\}$ |
| - | | @#FFBB00: $ ]a,b]\equiv\{x\ |\ \forall i.\ a_i<x_i\le b_i\}$ | | + | |
| - | | @#FFBB00: $ [a,b[\ \equiv\{x\ |\ \forall i.\ a_i\le x_i<b_i\}$ | | + | |
| - | | @#FFBB00: $ ]a,b[\ \equiv\{x\ |\ \forall i.\ a_i<x_i<b_i\}$ | | + | |
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Real_coordinate_space|Real coordinate space]] | Wikipedia: [[http://en.wikipedia.org/wiki/Real_coordinate_space|Real coordinate space]] | ||
| - | ==== Context ==== | + | ==== Parents ==== |
| === Subset of === | === Subset of === | ||
| [[Cartesian product]] | [[Cartesian product]] | ||
