Given two sets $A$ and $\varnothing$, where $\varnothing$ is the empty set. The Cartesian product is defined by
$$A \times \varnothing = \{ (x,y) \mid x \in A, y \in \varnothing \}$$
Why is $ A \times \varnothing = \varnothing$?
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1$\begingroup$ the question is whether you can exhibit an element in $A\times \emptyset$? What might such an element look like... $\endgroup$ – Ittay Weiss Sep 19 '14 at 0:36
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Because there are no ordered pairs, since there are no elements available for the second component.
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This is because for any $y$ in any universal set, $y \notin \emptyset$. It follows that even if $x \in A$, $(x,y) \notin A \times \emptyset$. The empty set has no elements.
