Relations and FunctionsIntroduction

Mathematics is the indispensable instrument of all physical research. – BERTHELOT

Riya is filling in a family worksheet for school. She writes: "Riya is the sister of Aman." True. Her little brother Aman copies the sentence the other way round: "Aman is the sister of Riya." Same two people, same words, and now the sentence is nonsense. The only thing that changed was the order.

Mathematics is full of sentences like this: 5 is less than 8, line l is parallel to line m, set A is a subset of set B. Each one links two objects, and to read it you have to know which object comes first. Such a link between objects is called a relation.

The swap test. Every sentence below is true. Swap its two objects — is it still true? Drag each sentence into the right bucket.

Only a few links — is parallel to, is a classmate of, is as tall as — survive the swap. Most of them care about which object comes first. So whenever mathematics links two objects, it writes them as a pair in a fixed order. In this chapter you will first build every possible pair from two sets, then pick out the pairs that obey a rule — a relation — and finally meet the most important relations of all, in which every input has exactly one output: functions. A function captures, with complete precision, how one quantity depends on another, and that is why almost every branch of mathematics and science runs on them.

Where the Word "Function" Came From

The idea of pairs is even older. In 1637 René Descartes showed how to describe every point of a plane by a pair of numbers. And in 1837 Peter Gustav Lejeune Dirichlet gave the modern definition you will learn in this chapter: a function is any rule that gives each input exactly one output.

Drag these milestones into the order in which they happened, earliest at the top.

Three hundred years of ideas — and the chapter starts with the oldest of them: pairs.