Cartesian product

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TODO: Find references

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Definition

Given two sets, $X$ and $Y$ their Cartesian product is the set:

• $X\times Y:=\{(x,y)\ \vert\ x\in X\wedge y\in Y\}$, note that $(x,y)$ is an ordered pair traditionally this means
  • $(x,y):=\{x,\{x,y\}\}$ or indeed
  • $X\times Y:=\Big\{\{x,\{x,y\}\}\ \vert\ x\in X\wedge y\in Y\Big\}$

Set construction

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TODO: Build a set that contains $\{x,y\}$s, then build another that contains ordered pairs, then the Cartesian product is a subset of this set

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