Concepts: sets, roster form and set-builder form

2.1 A set needs a rule you can test

The word set carries an everyday meaning that is too loose for mathematics. A cricket team and a crowd are collections, yet they are not sets in the technical sense until we can decide membership without argument. A collection becomes a set when every object either clearly belongs to it or clearly does not. This property is called being well-defined.

Compare two examples. The months of a year that begin with the letter J form a set, because you can check each month and answer yes or no. The ten most talented writers of India do not form a set, because talent is judged differently by different people, so the list is not fixed. The same objection rules out the eleven best batsmen of the world and the most dangerous animals.

Membership is written with the symbol in. If a is an element of A we write a in A, and if not we write a not in A. In the set of vowels V = {a, e, i, o, u}, a belongs to V while b does not. In the set of prime factors of 30, the number 3 belongs but 15 does not.

2.2 Roster form

In roster form, you list every element between braces and separate the entries with commas. The set of even positive integers below 7 is {2, 4, 6}. The vowels form {a, e, i, o, u}. The divisors of 42 form {1, 2, 3, 6, 7, 14, 21, 42}.

Three habits keep roster answers safe. Order does not matter, so {d, c, b, a} and {a, b, c, d} describe the same set. Repetition does not matter either, so the letters of SCHOOL give {S, C, H, O, L} with the repeated O written once. For an infinite set, write enough opening terms to show the pattern and close with three dots, as in {1, 3, 5, 7, and so on}.

Roster form has a limit. It cannot describe a set whose members follow no listable pattern. The set of real numbers is the standard example, because between any two reals there are more reals and no finite listing can begin to cover them.

2.3 Set-builder form

Set-builder form states the property that every member shares. The pattern is a variable, a colon, then the condition, all in braces. The colon means such that, and the braces mean the collection of everything that follows the rule.

An example makes the translation clear. The natural numbers strictly between 3 and 10 are 4, 5, 6, 7, 8 and 9, and the same collection can be written with a membership condition on x. Perfect squares need the rule x = n squared for some natural number n. A fraction pattern such as one half, two thirds, three quarters and so on needs a rule that links the numerator to the denominator.

Set-builder form is also the natural way to describe intervals of real numbers. The inequality -3 <= x < 5 describes a half-open interval, while x > 5 describes an unbounded interval. Moving between the inequality, the interval and the set-builder description is a skill examiners test directly, so practise all three directions.

2.4 The standard number sets

The chapter fixes symbols that return in every later chapter. N is the set of natural numbers, Z the integers, Q the rationals, R the reals and T the irrationals. The positive members are marked with a plus sign, so Z+ is the set of positive integers.

Their containment is a useful memory chain. Every natural number is an integer, so N lies inside Z. Every integer can be written with denominator 1, so Z lies inside Q. Every rational is real, so Q lies inside R. Irrational numbers are exactly the real numbers left outside Q, so T sits inside R and outside Q.

A common error is to treat zero as a natural number. In this book N starts at 1, which is why Z+ and N describe the same collection. Another error is to think pi is rational because it has a symbol; pi is irrational, and so are the square roots of primes.

2.5 Equal sets and repeated elements

Two sets are equal when they contain exactly the same elements, whatever order or repetition appears in the written form. {1, 2, 3} equals {2, 2, 1, 3, 3}, and the set of letters of ALLOY equals the set of letters of LOYAL once repeats are removed.

Equality is usually proved with two subset statements: show every element of A lies in B, then every element of B lies in A. For finite sets, comparing sorted rosters is quicker. For sets described by rules, solve the rule and compare the resulting elements.

A pair like {2, 3} and the solutions of x squared plus 5x plus 6 = 0 are not equal even though 2 and 3 both appear somewhere. The solutions of that equation are -2 and -3, because both signs fail in the original equation. Substituting the numbers back is the fastest way to catch this slip.

2.6 Where these two forms get tested

Exercise 1.1 asks for a decision on whether a collection is a set, then moves between roster form and set-builder form and asks you to list elements. The questions page adds matching exercises, where one set is written in roster form and the same set is described by a property on the other side.

Keep two checks before you finish any such question. First, confirm that the rule in set-builder form accepts every listed element and rejects everything outside the list. Second, confirm that the roster copy has no repeated element and no missing one. A single missing element changes the set, and a single rejected outsider makes the rule wrong.