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Relations and Functions
Introduction
Types of Relations
Practice: Relations and Equivalence Classes
Types of Functions
Practice: One-One and Onto Functions
Composition of Functions and Invertible Function
Miscellaneous Examples
Chapter Review: Mixed Practice
Summary
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Relations and FunctionsIntroduction

Reading time: ~15 min

The new session has just begun at a school in Delhi, and the sports teacher wants every Class XII student to coach one Class XI student before Sports Day. Before pairing anyone up, she notes down two facts about six of them: their age and the locality they live in.

Some pairings make sense for one reason, some for another. "The coach should be older." "They should live close by, so they can practise in the evening." Each reason picks out some of the possible pairs and leaves the rest.

You are the sports teacher. Call the Class XII students A and the Class XI students B. Each square of the grid is one pairing (a, b) with a from A and b from B — all nine squares together make the cartesian product A \times B. Choose a rule, then tap every square for which the rule is true.

Look at what you made. A rule in words — "is older than", "lives in the same locality as" — turned into a definite collection of ordered pairs chosen out of A \times B. Once the squares are marked, nobody needs the rule any more: the marked pairs say everything. That is exactly how mathematics thinks about a relation.

In Class XI you met relations and functions, together with their domain, co-domain and range, and the graphs of many real functions. The word "relation" comes from everyday English, where two objects are related if there is some recognisable link between them. With A the Class XII students and B the Class XI students of a school, each of these is a relation from A to B:

(i) \{(a, b) \in A \times B : a \text{ is a brother of } b\}

(ii) \{(a, b) \in A \times B : a \text{ is a sister of } b\}

(iii) \{(a, b) \in A \times B : \text{the age of } a \text{ is greater than the age of } b\}

(iv) \{(a, b) \in A \times B : a \text{ scored fewer marks than } b \text{ in the final examination}\}

(v) \{(a, b) \in A \times B : a \text{ lives in the same locality as } b\}

Strip away the story in each case and what remains is a collection of ordered pairs taken out of A \times B — just like your grid. So we define a relation R from A to B to be an arbitrary of A \times B. Any subset at all counts, even one that no sentence in words describes.

If (a, b) \in R, we say that a is related to b under R and write a R b. Once a pair is in R, we no longer ask why — belonging to R is the whole story. This is not a loss: a rule in words can be vague ("is a friend of"), while a set of pairs is perfectly precise.

As you saw in Class XI, functions are a special kind of relation: in a function, each element of the domain is related to exactly element of the co-domain.

In this chapter you will study the different types of relations and functions, the composition of functions and invertible functions. One colour code runs through every figure: the first set is blue, the second set is green, and the pairs or arrows that link them are purple. Red only ever marks a mistake or a missing pair.

Who Named the Function?

The idea of a function took about three hundred years to settle. René Descartes used the word in his book Géométrie (1637), but only for a positive whole-number power of a variable, while studying curves such as the parabola and the hyperbola. James Gregory (1667) called a function any quantity built from other quantities by repeated algebraic operations.

G. W. Leibniz at first used "function" (1673) for quantities that change from point to point on a curve — its coordinates, its slope, its tangent and normal. By 1714 he meant any quantity that depends on a variable, and he was the first to say "function of x".

Then came the notation. John Bernoulli wrote \varphi x for a function of x in 1718. Leonhard Euler (1734) made the letters f, F, \varphi, \psi the standard names for functions, and Joseph-Louis Lagrange (1793) wrote f(x), F(x) and \varphi(x) exactly as we do today. Lejeune Dirichlet gave the definition that was used until Georg Cantor built set theory; today's definition of a function as a special set of ordered pairs is simply Dirichlet's idea, made rigorous.

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

Euler makes f, F and φ the usual names for functions
Descartes uses the word "function" for powers of a variable
Dirichlet gives the definition used until set theory arrived
John Bernoulli writes φx for a function of x
Lagrange writes f(x), the notation we still use
Leibniz is the first to say "function of x"

From a word for powers of x to a set of ordered pairs — and the set-of-pairs idea is exactly where this chapter begins.

Sina