# Can you find any element that belongs to $X\times\emptyset$? [duplicate]

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?

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

$X\times \emptyset =\emptyset$
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.
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.