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My book says:

There is a unique function from $\emptyset$ to any set $A$.

I don't understand how that is. Let $A=\{1,2,3\}$. Which element of $A$ do we map $\emptyset$ to? Do we map $\emptyset$ to $\emptyset$? But $\emptyset\notin A$!

Thanks.

Martin Sleziak
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algebraically_speaking
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    There is but one way to nothing at all. – Olivier Bégassat Apr 17 '14 at 06:43
  • In order to understand the empty function, you must remember that a function in set theory is a set. Thus, the emptyset works well also as "emptyfunction" : just as the emptyset does not contain members, in the same way the "emptyfunction" does not map anything anywhere. – Mauro ALLEGRANZA Apr 17 '14 at 08:02

3 Answers3

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You don't map $\emptyset$ to an element of $A$ because $\emptyset$ is not an element of $\emptyset$. To define a map $\emptyset\to A$, we must, for each element of $\emptyset$, specify an element of $A$. This task is easy: Let's get started - done!

Hagen von Eitzen
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6

The function is called the empty function. The empty function $f:\emptyset \to A$ doesn't map $\emptyset$ to anything, since $\emptyset\notin \emptyset$. There is nothing in $\emptyset$, so $f$ doesn't map anything anywhere.

Why is it unqiue? Well, suppose we have a function $g:\emptyset\to A$. Then trivially $g(x)=f(x)$ for all $x\in \emptyset$ (since there are no such $x$), so $g=f$.

Alex Becker
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4

Recall that a function from $X$ to $Y$ is a subset of $X\times Y$ which has some particular properties.

If $X=\varnothing$, what is $X\times Y$? and how many subsets does it have?

Asaf Karagila
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