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I have a memory from years ago of someone saying that you can define zero to the power zero to be (either 1 or zero, I don't remember) by defining it as the number of mappings from the empty set to the empty set. I'd like to remember what he said and understand this approach.

I'm a high school maths teacher, who hasn't thought much about set theory since graduating with a maths degree 15 years ago, so I understand what the empty set is, but not much further.

Could anyone fill in the blanks for me on this explanation?

Many thanks in anticipation.

Andy876523
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    As to the rest of your question... the empty set is the unique set with zero elements, commonly notated ${}$ or $\emptyset$. As for relations, you can define a relation from $A$ to $B$ as being any subset of the cartesian product $A\times B$ (yes, even if $A$ or $B$ or both are the empty set) and functions are just specific types of relations. You have that there does in fact exist a relation from the empty set to itself... the empty relation (which happens to be equal to the empty set). – JMoravitz May 30 '20 at 19:13

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In set theory a function from $A$ to $B$ is defined as a subset of $A\times B$ such that every element of $A$ appears in exactly one pair in the subset. $\emptyset \subseteq \emptyset \times \emptyset$ vacously satisfies this (since $\emptyset$ has no elements). So there is exactly one function from $\emptyset$ to itself. So if you want the number of functions from $A$ to $B$ to be $|B|^{|A|}$ you get $0^0 = 1$.

BrianO
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Jacob FG
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