Differences
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real_coordinate_space [2014/03/21 11:11] 127.0.0.1 external edit |
real_coordinate_space [2014/04/04 16:01] nikolaj |
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| ==== Set ==== | ==== Set ==== | ||
| | @#55CCEE: context | @#55CCEE: $ n\in \mathbb N $ | | | @#55CCEE: context | @#55CCEE: $ n\in \mathbb N $ | | ||
| - | |||
| | @#FFBB00: definiendum | @#FFBB00: $\mathbb R^n$ | | | @#FFBB00: definiendum | @#FFBB00: $\mathbb R^n$ | | ||
| - | Recursive definition: | + | ==== Discussion ==== |
| + | Explicitly, | ||
| - | | $ p\in \mathbb N $ | $ p>1 $ | | + | $ \mathbb R^1=\mathbb R $ |
| + | $ \mathbb R^p=\mathbb R^{p-1}\times \mathbb R, \hspace{1cm}$ | ||
| - | | @#55EE55: postulate | @#55EE55: $ \mathbb R^1=\mathbb R $ | | ||
| - | | @#55EE55: postulate | @#55EE55: $ \mathbb R^p=\mathbb R^{p-1}\times \mathbb R, \hspace{1cm}$ | | ||
| - | |||
| - | ==== Discussion ==== | ||
| | @#DDDDDD: range | @#DDDDDD: $1\le i\le p$ | | | @#DDDDDD: range | @#DDDDDD: $1\le i\le p$ | | ||
| Line 20: | Line 17: | ||
| | @#EEEE55: predicate | @#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: definiendum | @#FFBB00: $ [a,b]\equiv\{x\ |\ \forall i.\ a_i\le x_i\le b_i\}$ | | + | >todo: define the following in a seperate entry: |
| - | | @#FFBB00: definiendum | @#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\le x_i<b_i\}$ | | + | $ [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<x_i<b_i\}$ | | + | |
| + | $ ]a,b]\equiv\{x\ |\ \forall i.\ a_i<x_i\le b_i\}$ | ||
| + | |||
| + | $ [a,b[\ \equiv\{x\ |\ \forall i.\ a_i\le x_i<b_i\}$ | ||
| + | |||
| + | $ ]a,b[\ \equiv\{x\ |\ \forall i.\ a_i<x_i<b_i\}$ | ||
| === Reference === | === Reference === | ||
| Line 31: | Line 33: | ||
| === Subset of === | === Subset of === | ||
| [[Cartesian product]] | [[Cartesian product]] | ||
| - | === Context === | + | === Requirements === |
| [[Real number]] | [[Real number]] | ||
