The set $B^{A}$ of all functions $A \to B$

Let $A$ and $B$ be sets. By For sets $A$ and $B$ the collection of all functions $A \to B$ is a set, being a subset of $\mathcal{P}(A \times B)$ the functions $A \to B$ (A function is a relation $f$ with $(a,b) \in f$ and $(a,c) \in f$ implying $b = c$; $f : A \to B$, the value $f(a)$, domain and codomain) form a set; it is written

$B^{A} := \{\, f : f \text{ is a function with } \operatorname{dom} f = A \text{ and } \operatorname{ran} f \subseteq B \,\}.$

Thus $f \in B^{A}$ holds if and only if $f : A \to B$.

• The notation collides with exponentiation, and the collision is deliberate. $B^{A}$ is the same symbol the library already uses for a space of functions in The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = \{0, 1, \dots, n-1\}$ ↗, where $F^{X}$ is the set of functions $X \to F$ carrying a vector-space structure; the underlying set there is the one defined here. Some set-theory texts write ${}^{A}B$ instead, reserving $B^{A}$ for ordinal and cardinal exponentiation. This library keeps $B^{A}$ for the function set, and Ordinal $\alpha^{\beta}$ and cardinal $\kappa^{\lambda}$ are different operations that share one notation ↗ is where the arithmetic exponentiations are distinguished from one another.

• Degenerate cases. $B^{\varnothing}$ has exactly one element, the empty function, for every $B$, since a function with empty domain has no elements at all. $\varnothing^{A}$ is empty whenever $A$ is nonempty, since a function with domain $A$ must take a value at each point of $A$, and $\varnothing^{\varnothing}$ again has the empty function as its only element.