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If you have two sets $A$ and $B$ where $A=B=\{\;\}$, then what would $A\times B$ be equal to?

I know that $A\times B=\{(a,b)|a\text{ is an element of }A\text{ and }b\text{ is an element of }B\}$.

I think I am getting confused as to the notation of an element of an empty set or even if you can have one.

lioness99a
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W. G.
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  • The final set should contain pair of the form $(a,b)$. But clearly there can be no pair, hence the final set is empty. – Zubzub Apr 07 '17 at 12:56
  • "an element of an empty set or even if you can have one"... well, of course you can't have one. Otherwise it wouldn't be an empty set, it would be a set with an element! – 5xum Apr 07 '17 at 13:02

1 Answers1

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Keep calm when you get confused, just apply the definition.

$A\times B$ should consist of exactly those $(a,b)$ such that $a\in A$ and $b\in B$. Now if any of $A$ or $B$ is empty that simply doesn't happen and therefore there is no $(a,b)$ that qualifies for being a member of $A\times B$, that is there's no member of $A\times B$. This means that we always have

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

for any set $X$. Especially we get:

$$\emptyset\times\emptyset = \emptyset$$

skyking
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  • Does {}={( , )} then? That helps me thank you! – W. G. Apr 07 '17 at 13:03
  • @W.G. No, the empty set does not contain anything whatsoever. You use the definition to see that the product set does not contain anything at all, that is it's the empty set. – skyking Apr 07 '17 at 13:10