Differences
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total_order [2013/09/08 14:57] nikolaj |
total_order [2014/03/21 11:11] (current) |
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| ===== Total order ===== | ===== Total order ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $X$ | | + | | @#55CCEE: context | @#55CCEE: $X$ | |
| - | | @#FFBB00: $ \le\ \in\ \mathrm{it} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ \le\ \in\ \mathrm{it} $ | |
| The relation $\le$ is an order relation if it's in the intersection of all total, all anti-symmetric and all transitive relation. Hence | The relation $\le$ is an order relation if it's in the intersection of all total, all anti-symmetric and all transitive relation. Hence | ||
| - | | @#88DDEE: $ \le\ \in\ \mathrm{Rel}(X) $ | | + | | @#55CCEE: context | @#55CCEE: $ \le\ \in\ \mathrm{Rel}(X) $ | |
| | $ x,y,z \in X $ | | | $ x,y,z \in X $ | | ||
| - | | @#55EE55: $ x \le y\ \lor\ y \le x $ | | + | | @#55EE55: postulate | @#55EE55: $ x \le y\ \lor\ y \le x $ | |
| - | | @#55EE55: $ x\le y\ \land\ y\le x \implies (x=y) $ | | + | | @#55EE55: postulate | @#55EE55: $ x\le y\ \land\ y\le x \implies (x=y) $ | |
| - | | @#55EE55: $ x \le y\ \land\ y \le z \Leftrightarrow x\le z $ | | + | | @#55EE55: postulate | @#55EE55: $ x \le y\ \land\ y \le z \Leftrightarrow x\le z $ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
