What is a set?

A set is a well-defined collection of objects. 'Well-defined' means that anyone, anywhere, can decide for sure whether a given object belongs to it or not.

• 'The odd numbers less than 15', 'the vowels of English' and 'the prime numbers less than 20' are sets: everyone gets the same members.
• 'The five best cricketers in India' is not a set: your list and mine would differ, because 'best' is a matter of opinion.

The objects in a set are its elements (or members). We name sets with capital letters, and write a ∈ A for 'a belongs to A' and b ∉ A for 'b does not belong to A'. If A = {2, 3, 5, 7}, then 5 ∈ A but 9 ∉ A.

There are two ways to write a set:

• Roster form: list the elements inside braces { }, separated by commas. Order does not matter, and each element is written once: the letters of SCHOOL form {S, C, H, O, L}.
• Set-builder form: describe the property the elements share, {x : property of x}, read 'the set of all x such that ...'.

Going from roster to set-builder form means spotting the pattern:

• {1, 2, 3, 6, 9, 18} = {x : x is a natural number that divides 18}.
• {2, 4, 8, 16, 32} = {x : x = 2ⁿ, n ∈ N, 1 ≤ n ≤ 5}.
• {1/2, 2/3, 3/4, ..., 7/8} = {x : x = n/(n + 1), n ∈ N, 1 ≤ n ≤ 7}.

Always say what kind of number n is. Writing only '1 ≤ n ≤ 5' would allow n = 1.5 too; adding n ∈ N limits it to whole counting numbers.

Some sets have standard names: N (natural numbers), Z (integers), Q (rational numbers), R (real numbers). Z⁺ means the positive integers, and so on.