Draft:Set exponentiation

In set theory, the set exponentiation of $Y$ to the $X$, is the set of all functions from $X$ to $Y$, denoted $Y^{X}$.

Definition

- Empty function

- Formal definition

$Y^{X}:=\{f\subseteq (X\times Y)\,|\,\forall x\in X,\;\exists !y\in Y\,((x,y)\in f)\}$

Note that, often, $f:A\mapsto B\;:\equiv \;f\in B^{A}$

Natural-number powers (Aⁿ)

Set of functions from $\{1,2,\cdots ,n\}$ to $A$.

Cardinality

$|B^{A}|=|B|^{|A|}$

See Suppes and Kuratowski (p. 170):

$(A^{B})^{C}\approx A^{B\times C}$

$A^{B\,\sqcup \,C}\approx A^{B}\times A^{C}$

$(A\times B)^{C}\approx A^{C}\times B^{C}$

Applications

- Curring

History

https://hsm.stackexchange.com/questions/11586/

References

• McCarty, George (1988) [1967]. Topology: An Introduction with Application to Topological Groups. 9: Dover. ISBN 978-0486656335. LCCN 87-34583. Archived from the original on 2019-07-04.{{cite book}}: CS1 maint: location (link)
• Stoll, Robert R. (1963). Set Theory and Logic. San Francisco: W. H. Freeman. ISBN 7167 0416-1. LCCN 63-8995. {{cite book}}: ISBN / Date incompatibility (help)
• Suppes, Patrick (1972) [1960]. Axiomatic Set Theory. Dover Books on Mathematics. New York: Dover. ISBN 0-486-61630-4. LCCN 72-86226. Archived from the original on 2014-08-06.
• Takeuti, Gaisi; Zaring, Wilson M (1982). Introduction to Axiomatic Set Theory. Graduate Texts in Mathematics (2nd ed.). New York: Springer-Verlag. doi:10.1007/978-1-4613-8168-6. ISBN 0-387-90683-5. ISSN 0072-5285. LCCN 81-8838. Archived from the original on 2014-08-06.
• Kuratowski, Kazimierz (1968). Set Theory. Amsterdam: North Holland Publishing. LCCN 67-21972.