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real_coordinate_space [2013/09/03 16:28] nikolaj |
real_coordinate_space [2014/03/21 11:11] 127.0.0.1 external edit |
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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: $\mathbb R^n$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\mathbb R^n$ | |
| Recursive definition: | Recursive definition: | ||
| Line 9: | Line 9: | ||
| | $ p\in \mathbb N $ | $ p>1 $ | | | $ p\in \mathbb N $ | $ p>1 $ | | ||
| - | | @#55EE55: $ \mathbb R^1=\mathbb R $ | | + | | @#55EE55: postulate | @#55EE55: $ \mathbb R^1=\mathbb R $ | |
| - | | @#55EE55: $ \mathbb R^p=\mathbb R^{p-1}\times \mathbb R, \hspace{1cm}$ | | + | | @#55EE55: postulate | @#55EE55: $ \mathbb R^p=\mathbb R^{p-1}\times \mathbb R, \hspace{1cm}$ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| - | | @#DDDDDD: $1\le i\le p$ | | + | | @#DDDDDD: range | @#DDDDDD: $1\le i\le p$ | |
| - | | @#EEEE55: $ a< b \equiv \forall i.\ a_i< b_i$ | | + | | @#EEEE55: predicate | @#EEEE55: $ a< b \equiv \forall i.\ a_i< b_i$ | |
| - | | @#EEEE55: $ a> b \equiv \forall i.\ a_i> b_i$ | | + | | @#EEEE55: predicate | @#EEEE55: $ a> b \equiv \forall i.\ a_i> b_i$ | |
| - | | @#EEEE55: $ a\le b \equiv \forall i.\ a_i\le b_i$ | | + | | @#EEEE55: predicate | @#EEEE55: $ a\le b \equiv \forall i.\ a_i\le b_i$ | |
| - | | @#EEEE55: $ a\ge b \equiv \forall i.\ a_i\ge b_i$ | | + | | @#EEEE55: predicate | @#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\}$ | | + | | @#FFBB00: definiendum | @#FFBB00: $ [a,b]\equiv\{x\ |\ \forall i.\ a_i\le x_i\le b_i\}$ | |
| - | | @#FFBB00: $ ]a,b]\equiv\{x\ |\ \forall i.\ a_i<x_i\le b_i\}$ | | + | | @#FFBB00: definiendum | @#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: definiendum | @#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\}$ | | + | | @#FFBB00: definiendum | @#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 ==== |
| - | === Refinement of === | + | === Subset of === |
| [[Cartesian product]] | [[Cartesian product]] | ||
| - | === Requirements === | + | === Context === |
| [[Real number]] | [[Real number]] | ||
