Cartesian products and relations

Functions and relations

The clarification of the concept of function, even when confined to the subject of analysis, had become a matter of desperate importance in the eighteenth and nineteenth centuries. Fourier's representation of piecewise continuous functions [a, b] → ℝ by trigonometric series and, later, Weierstrass and Riemann's examples of continuous nowhere differentiable functions and functions which are nowhere continuous yet integrable forced a generation of mathematicians (who were accustomed to treating functions of real variables as visualizable geometric entities) to reluctantly revise their understanding. A precise and general definition of function which is not tied to analysis is expressible within the set-theoretic framework.

Functions and relations saturate every part of mathematics. Even in our humble introduction we had to introduce these ideas at an early stage in order to construct models of the basic number systems. Now that the logical prerequisites have been brought into focus it is worth reviewing the formal definitions.