# Does there exist a bijection between empty sets?

Does there exist a bijection between empty sets?

What I think is:

Since 'for every $x\in \emptyset$, there exists $y\in \emptyset$ such that $(x,y)\in f$' is false and the negation of this statement is also false because of 'existence' sentence..

But I think empty sets are equipotent intuitively.
Am I wrong? Help

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Try to use $f=\emptyset$. Is this a function from $\emptyset$ to $\emptyset$? Is it injective/surjective/bijective? (Just try to write down the definitions, it should be easy.) \\ If you still have problems, maybe it would be useful if you have a look at empty function and vacuous truth at Wikipedia. –  Martin Sleziak May 4 '12 at 13:04
"For every $x$ in $\varnothing$, ..." is always vacuously true. Kind of like being innocent until proven guilty. –  Rahul May 4 '12 at 13:13
There is only one empty set. So they're clearly in bijection. –  lhf May 4 '12 at 13:20
@rahul 'for every x,y in $\emptyset$ , $\emptyset(x) is not equal to \emptyset(y)$' is true then? –  Katlus May 4 '12 at 13:29
@Katlus Every claim which has the form $(\forall x\in\emptyset) \varphi(x)$ is true; where $\varphi(x)$ can be arbitrary formula. The claim that you wrote in your comment has this form. –  Martin Sleziak May 4 '12 at 13:32
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Yes, there is.

(As said in the comments) the function $f=\emptyset$ is the bijection you seek.

It may seem strange that $f$ is a set, but this is ok since the definition of a function is a set of pairs (s.t ...), in this case we take $f$ to be an empty set.

The definition of $f$ is fine since there are no $x, y_1,y_2 \in \emptyset$ s.t $y_1\neq y_2$ and s.t $(x,y_1),(x,y_2)\in f$.

$f$ is injective since there are no two different elements in $\emptyset$ that $f$ maps to the the same element (this is just because there are no two different elements in $\emptyset$).

$f$ is surjective since for each $y$ in $\emptyset$ there exist a source (this is since there are no elements in $\emptyset$ hence we can't say there exist an element that doesn't have a source).

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If a statement starts with '$\forall x \in \emptyset$,...', it is trivially true.

Because the set contains no elements, any statement holds for 'all elements' in it.

I'm not sure if this answer is 'mathematical' enough, but I think that's the point you might have missed.

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The term is "vacuously true" is more common (and has a clearer meaning, while "trivially true" implies just that the argument is "trivial"). –  Asaf Karagila May 4 '12 at 13:35
@Katlus: What? How did you derive that from my answer? There is a bijection between the empty set and itself. Note that any bijection between two sets preserves cardinality and the only set of cardinality zero is the empty set. If there exists a bijection between $A$ and the empty set it has cardinality zero and therefore empty. –  Asaf Karagila May 4 '12 at 13:37