Sets and their representations

A set is a well-defined collection of objects: for every object it must be possible to decide, with no room for opinion, whether it belongs. The rivers of India form a set; the five most talented batsmen do not, because talent is a judgement.

The objects in a set are its elements or members. Sets are named with capital letters and elements with small letters; a ∈ A is read "a belongs to A" and b ∉ A is read "b does not belong to A".

Roster (tabular) form lists the elements inside braces, separated by commas: the divisors of 12 are {1, 2, 3, 4, 6, 12}. The order of listing does not matter, and a repeated element is written only once, so the letters of CRICKET give {C, R, I, K, E, T}.

Set-builder form states one property that every element has and nothing outside the set has: {x : x ∈ N, 3 < x < 10} stands for {4, 5, 6, 7, 8, 9}. The braces are read "the set of all" and the colon "such that".

An infinite set with a visible pattern can be written in roster form with three dots, such as {1, 3, 5, …} for the odd natural numbers; a set like R, whose elements follow no listable pattern, cannot be written in roster form at all.

Standard names used throughout: N (natural numbers), Z (integers), Q (rational numbers), R (real numbers), and Z⁺, Q⁺, R⁺ for the positive integers, positive rationals and positive reals.

Switching between forms is routine: the solution set of x² − 7x + 12 = 0 in roster form is {3, 4}, and {2, 4, 8, 16, …} in set-builder form is {x : x = 2ⁿ, n ∈ N}.