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While explaining that "In the category of sets the Cartesian product $\prod_{i \in I} A_{i}$ is a product of the family $\{A_{i}: i \in I\}$."

After showing this the author gave the following remark :

enter image description here

But I did not understand why if some $A_{j} = \emptyset,$ then the whole $\prod_{i \in I} A_{i} = \emptyset$ and why there can be no function that satisfies what given in the picture, is not there a mistake in the index j? it must be i instead? could anyone explain this for me please?

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$j$ is a particular index for which $A_j$ is the empty set.

Consider an element $x$ in the product. $x$ has a component $x_i$ for each $i\in I$. When it comes to the $j^{th}$ component $x_j$, what can it be? There aren't any options in $A_j$, which is empty, so $x$ can't exist and the product must be empty too.

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  • $\begingroup$ $x_{j}$ can not be zero in this case? $\endgroup$ – user426277 Nov 24 '17 at 14:37
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    $\begingroup$ @Idonotknow Well zero is not an element of the empty set. $\endgroup$ – Mark Bennet Nov 24 '17 at 14:47

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