Chapter 10 Functions
One of the reasons to introduce relations is because they are nice intermediate mathematical object between sets (which have very little additional structure) and functions (which have quite a lot of structure). Indeed the idea of relations allows us to escape from the idea of a function as being a formula. Arguably, the usual high-school mathematics curriculum (especially the last couple of years of it) is really driving us towards being able to do calculus. And in calculus all the functions we look at are nice formulas that build up more complicated functions by doing arithmetic on simpler functions. At their core, these functions are really very algorithmic:
- Give me an input number \(x\)
- I do some arithmetic on \(x\text{,}\) and maybe look up some values (in a table or via a calculator or computer 128 ) of things like sine, or logarithms.
- Then I return to you numerical result \(y\text{.}\)
Of course to be a function, this procedure has to be well defined — if you give me one input then I return to you one output. And if you give me the same input twice then I’d better return the same output each time.
Since doing complicated mathematical computations by hand can be very laborious, people who needed the results of those computations would hire people to do those computations for them. These people were called computers.
