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cartesian_product [2014/04/01 16:26] nikolaj |
cartesian_product [2014/04/06 19:10] nikolaj |
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| ===== Cartesian product ===== | ===== Cartesian product ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#55CCEE: context | @#55CCEE: $X,Y$ | | + | | @#55CCEE: context | @#55CCEE: $X,Y$ ... small set | |
| | @#FFBB00: definiendum | @#FFBB00: $ p\in X \times Y $ | | | @#FFBB00: definiendum | @#FFBB00: $ p\in X \times Y $ | | ||
| - | | $ x\in X$ | | + | | @#DDDDDD: range | @#DDDDDD: $ x\in X$ | |
| - | | $ y\in Y$ | | + | | @#DDDDDD: range | @#DDDDDD: $ y\in Y$ | |
| - | | @#55EE55: postulate | @#55EE55: $ \exists x.\ \exists y.\ p=\langle x,y\rangle $ | | + | | @#55EE55: postulate | @#55EE55: $ \exists x,y.\,p=\langle x,y\rangle $ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| - | | @#55CCEE: context | @#55CCEE: $ X \times Y\subset \mathcal{P}(\mathcal{P}(X \cap Y))$ | | + | ^ $ X \times Y\subset \mathcal{P}(\mathcal{P}(X \cap Y))$ ^ |
| === Definitions === | === Definitions === | ||
| In accordance to the defintion of the n-tuple in [[Ordered pair]], we set | In accordance to the defintion of the n-tuple in [[Ordered pair]], we set | ||
| - | | @#FFBB00: definiendum | @#FFBB00: $ X_1\times X_2\times X_3 \equiv (X_1\times X_2)\times X_3 $ | | + | ^ $ X_1\times X_2\times X_3 \equiv (X_1\times X_2)\times X_3 $ ^ |
| and inductively for | and inductively for | ||
| - | | @#FFBB00: definiendum | @#FFBB00: $ X_1\times X_1\times X_3\times \ \dots\ \times X_{n-1}\times X_n \equiv ((\dots((X_1\times X_2)\times X_3)\times\ \dots\ )\times X_{n-1})\times X_n $ | | + | ^ $ X_1\times X_1\times X_3\times \ \dots\ \times X_{n-1}\times X_n \equiv ((\dots((X_1\times X_2)\times X_3)\times\ \dots\ )\times X_{n-1})\times X_n $ ^ |
| ==== Parents ==== | ==== Parents ==== | ||
| === Requirements === | === Requirements === | ||
| [[Ordered pair]] | [[Ordered pair]] | ||
| - | + | === Related === | |
| + | [[Product type]] | ||
