Can a function map into or out of the null set?

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A function can map out of a null set, but this action is effectively meaningless, similar to the identity function. It can also map into a null set, but only if the domain contains no elements, as any non-empty domain would require a mapping to a value. Both scenarios result in the empty function, which is the central point of the discussion. The empty function serves as the only valid example for these mappings. Thus, the concept of mapping into or out of a null set ultimately leads to the empty function.
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Can a function involve a mapping into or out of a null set?
 
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Gear300 said:
Can a function involve a mapping into or out of a null set?

Out of: yes, but it doesn't do anything (in an even more literal sense than the identity function doesn't do anything).

Into: Yes, but if it's a function rather than a partial function the domain must have exactly zero elements (because if it had even one it would have needed to map that to something).

So in both cases it's just the empty function, as jbunniii points out.
 

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