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In "Introduction to Set Theory" we have the following:

Theorem Let $f$ and $g$ be functions. Then $g \circ f$ is a function. $g \circ f$ is defined at $x$ if and only if $f$ is defined at $x$ and $g$ is defined at $f(x)$, i.e., $$ \text{dom }(g \circ f)= \text{dom } f \cap f^{-1}[\text{dom }g].$$

This assertion is certainly correct, but isn't it redundant to cite $\text{dom } f$? That is, isn't $f^{-1}[\text{dom }g]$ a subset of $\text{dom }f$?

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    Yes, but it is spelling out what the intuitive version of the theorem says for clarity. They should clarify the redundancy though. – not all wrong Jul 16 '13 at 01:53

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Yes, it is. By definition

$$f^{-1}[\operatorname{dom}\,g]=\{x\in\operatorname{dom}f:f(x)\in\operatorname{dom}\,g\}\;.$$

Brian M. Scott
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