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Why the cardinality of $$\{f:\emptyset\to \emptyset \mid f\text{ a function}\}$$ is one ? There is no such function... the cardinality should be $0$, no ? How could a function $$f:\emptyset \to \emptyset$$ looks like ?

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$\emptyset$ is the one and only function $\emptyset\to\emptyset$. As weird as it may sound in laymen's terms, you can see for yourself that:

  • it is a (the only) subset of $\emptyset\times \emptyset$
  • for all $a\in\emptyset$ there is some $b\in\emptyset$ such that $(a,b)\in\emptyset$
  • for all $(a_1,b_1),(a_2,b_2)\in\emptyset$, either $a_1\ne a_2$ or $b_1= b_2$.

Additional exercise: What happens if we consider functions $\emptyset\to A$ for $A\ne\emptyset$? And if we consider functions $B\to\emptyset$ with $B\ne \emptyset$?

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