ABSTRACT
The concept of map (or function) has a long history. Originally functions were understood to be given by more or less explicit “formulae” (polynomial, rational, algebraic, and later by series). Controversies around what the “most general” functions should be arose, for instance, in connection with solving partial differential equations (by means of trigonometric series); this is somewhat parallel to the controversy around what the “most general” numbers should be that arose in connection with solving algebraic equations (such as x2 = 2, x2 = −1, or higher degree equations with no solutions expressed by radicals, etc.). The notion of “completely arbitrary” function gradually arose through the work of Dirichlet, Riemann, Weierstrass, Cantor, etc. Here is the definition:
Definition 14.1. A map (or function) from a set A to a set B is a subset F ⊂ A × B such that for any a ∈ A there is a unique b ∈ B with (a, b) ∈ F . If (a, b) ∈ F we write F (a) = b or a 7→ b or a 7→ F (a). We also write F : A → B or A
F→ B. Remark 14.2. The above defines a new (ternary) relational predicate µ equal
to “...is a map from ... to ....” Also we may introduce a new functional symbol F̂ by
∀x∀y((µ(F,A,B) ∧ (x ∈ A) ∧ (y ∈ B))→ ((F̂ (x) = y)↔ ((x, y) ∈ F ))). Here F,A,B can be constants or variables depending on the context. We will usually drop the ̂ (or think of the argot translation as dropping the hat). Also note that what we call a map F ⊂ A × B corresponds to what in elementary mathematics is called the graph of a map.
