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So for example, $(0.2,0.8)\times(0.5,0.7)$ as a subset of $\mathbb{R}^2$ is defined to be the set $\{\langle x,y\rangle\mid 0.2<x<0.8$ and $0.5<y<0.7\}$, which essentially looks like a box.

But what if one of the factor sets is the empty set? Would that equate to $\emptyset$ in $\mathbb{R}^2$?

Is the statement $(0.2,0.8)\times\emptyset = \emptyset\times\emptyset=\emptyset$ be correct?

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  • $\begingroup$ 0 in RxR? Of course not. The empty set is not an element of RxR. $\endgroup$ – William Elliot Oct 6 '17 at 2:14
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from definition:

$$A \times B = \{ (a, b): a \in A, b \in B \}$$

Hence

$$A \times \emptyset = \{ (a, b): a \in A, b \in \emptyset \} = \emptyset$$

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