let $X$ and $Y$ be sets and $Y^X$ the set of function $f:X\to Y$. How can we interpret $Y^X$ as the cartesian product $\prod_{x\in X}Y_x$ where $Y_x=Y$ for each $x\in X$?
Another interpretation of function space
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elementary-set-theory
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0They are the same. No interpretation is needed. – 2012-02-23
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1I’m not really sure what your question is: by definition that Cartesian product is the set of functions from $X$ to $Y$. – 2012-02-23
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0for example suppose $X$ is finite. then we have the bijection $$\prod_{x\in X}Y_x\to Y^X$$ defined by sending a tuple $$(y_1,y_2,...,y_n)$$ maps to the map $f$ that sends $f(x_1)=y_1,...,f(x_n)=y_n$ is that correct? – 2012-02-23
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2Yes, that is correct. – 2012-02-23