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I meet with term 'empty function' from time to time. It's high time to understand its nature. What is field( set of arguments) and what is image? ( set of value)?

user180834
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The set theoretic definition of a function $f:A\rightarrow B$ is a special type of binary relation. I.e. it is a subset of $A \times B$. The empty set is also a subset of $A \times B$ and is defined as the empty function.

Curious Droid
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    This correctly defines "empty relation", not empty function. A function needs to be defined, uniquely, on each element of the domain. – vadim123 Oct 20 '14 at 16:25
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    Yeah, specifically the empty relation is a function iff $A$ is empty. – Seth Oct 20 '14 at 16:28