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Note that $X\times Y=${$(x,y):x\in X$ and $y\in Y$} and $(x,y)=${{$x$},{$x,y$}}.

Find $X\times\emptyset$.

How can we use definitions for the question?

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  • $\begingroup$ What does mean "find"? $\endgroup$ – Marco Lecci Nov 2 '16 at 17:47
  • $\begingroup$ Can you find any element that belongs to $X\times\varnothing$? $\endgroup$ – drhab Nov 2 '16 at 17:48
  • $\begingroup$ @MarcoLecci Can you find any element that belongs to $X\times\emptyset$? $\endgroup$ – James Ensor Nov 2 '16 at 17:50
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$X\times \emptyset =\emptyset $

We do this by contradiction. Suppose $X\times \emptyset \ne\emptyset $. Then $\exists (x_1,x_2)\in X\times \emptyset .$

This implies $x_2\in \emptyset ,$ which is a contradiction.

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Since there are no elements in $\emptyset$, there are no pairs $(x,y)$ with $x \in X$ and $y \in \emptyset$.

Therefore $X \times \emptyset$ is empty.

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