Chapter 2.1 — Introduction to Sets
Introduction of sets, roster form and set builder form. This is Lesson 1 of 5 in Chapter 2: Sets.
A Rule You Can Always Check
A set is a well-defined collection of distinct objects, called its elements. "Well-defined" is the part that matters: given any object, you must be able to say for certain whether it belongs to the collection or not. "Tall students in a class" fails that test — tallness is a matter of opinion — while "students taller than 160 cm" passes it instantly, since height is something you can measure and check.
What Doesn't Qualify as a Set
It's just as useful to see collections that fail the well-defined test as ones that pass it. "The most beautiful paintings in a museum" isn't a set — two visitors could reasonably disagree about which paintings count. Neither is "the difficult questions in an exam," since difficulty depends entirely on who's answering. Compare these to "the paintings in a museum painted before 1900" or "the questions in an exam worth more than 5 marks" — both instantly well-defined, because a date and a mark value can be checked without anyone's opinion entering into it.
The pattern to look for is whether the rule involves a measurable fact (a date, a count, a mathematical property) or a judgment call (beauty, difficulty, talent). Only the first kind produces a genuine set — and this single distinction is the one test that decides whether any collection you encounter, in this chapter or beyond it, is actually eligible to be written with set notation at all.
Naming and Writing a Set
Sets are written by listing their elements inside a pair of braces — also called curly brackets — and are usually named with a capital letter.
| Set | Description | Written as |
|---|---|---|
| N | Natural numbers | N = {1, 2, 3, 4, 5, …} |
| M | Letters in the word "MATHEMATICS" | M = {M, A, T, H, E, I, C, S} |
| P | Planets of the solar system | P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} |
| V | Vowels | V = {a, e, i, o, u} |
| C | First five cubic numbers | C = {1, 8, 27, 64, 125} |
Notice that "MATHEMATICS" contributes only eight letters to set M, not eleven — a set never lists an element more than once, however many times it repeats in the original word. The same rule applies to the planets set P: it has eight elements because there are eight distinct planets, regardless of how the collection might be described or reordered.
Belongs To, or Doesn't
Once a set is written down, you can ask whether any particular object is one of its elements. If x is present in set A, this is written x ∈ A and read "x belongs to A." If x is not present, it's written x ∉ A.
x ∈ A means "x belongs to A" x ∉ A means "x does not belong to A"- 3 ∈ N, but 0 ∉ N (natural numbers start at 1)
- a ∈ M, but l ∉ M (using the letter set for "MATHEMATICS" above)
- e ∈ V, but g ∉ V
- 8 ∈ C, but 31 ∉ C
Two Ways to Describe the Same Set
Every set can be written in either of two equivalent forms: roster form, which lists every element directly, or set-builder form, which describes elements through a shared property instead of naming them one by one.
| Roster form | Set-builder form |
|---|---|
| R = {1, 2, 3, 4, 5} | R = {x : x ∈ N, x ≤ 5} |
| A = {a, e, i, o, u} | A = {x : x is a vowel in the English alphabet} |
| M = {3, 6, 9, 12, 15, 18} | M = {x : x is a multiple of 3 and x < 20} |
| S = {−2, −1, 0, 1, 2} | S = {x : x ∈ Z, −2 ≤ x ≤ 2} |
| C = {2, 5, 10, 17} | C = {x : x = n² + 1, n ∈ N, n ≤ 4} |
The last row is worth checking by hand: substituting n = 1, 2, 3, 4 into n² + 1 gives 2, 5, 10, 17 — exactly the roster form on the left. Whenever a set-builder rule involves a formula rather than a simple property, generating the roster form is just a matter of plugging in every allowed value of the variable.
Turning a Description Into a List
Converting a plain-English or set-builder description into roster form is mostly a matter of checking every candidate carefully, without missing one or including one that doesn't qualify.
- G = all the factors of 20 — check every number from 1 to 20 for exact division: G = {1, 2, 4, 5, 10, 20}
- F = multiples of 4 between 17 and 61, divisible by 7 — multiples of 4 in that range are 20, 24, 28, …, 60; of those, only 28 and 56 are also divisible by 7: F = {28, 56}
- S = the letters in the word "MADAM" — M, A, D, A, M has only three distinct letters: S = {M, A, D}
- P = whole numbers between 3.5 and 6.7 — the whole numbers strictly between those two decimals are: P = {4, 5, 6}
Three More Descriptions, Worked Through
The same care applies whether the description involves numbers, words, or everyday categories.
- B = all months of the year with exactly 30 days — checking each month against the calendar: B = {April, June, September, November}
- P = all prime numbers smaller than 10 — testing 2 through 9 for exactly two factors: P = {2, 3, 5, 7}
- X = the colours of the rainbow — X = {violet, indigo, blue, green, yellow, orange, red}
A factors-of-12 question makes a good final check on this skill: the factors of 12 are 1, 2, 3, 4, 6, and 12 — so out of the candidates 1, 4, 5, and 12, only 5 is not a member, since 12 is not divisible by 5 at all.
One More Skill: Matching a List to Its Rule
The reverse direction — given a roster form, finding the rule that generated it — comes up just as often. Given {p, r, i, n, c, a, l}, the rule is simply "letters of the word PRINCIPAL." Given {1, 2, 3, 6, 9, 18}, checking each number against 18 shows all six divide it evenly, so the rule is "positive divisors of 18." Given {3, −3}, both numbers square to 9, so the rule is "integers x such that x² − 9 = 0." And given {0}, the single element satisfies x + 1 = 1 — the rule doesn't need to look complicated to be correct, only accurate.
Where This Chapter Goes Next
This introduction covers the vocabulary — well-defined collections, membership notation, and the two ways of writing a set — that every later exercise in this chapter assumes. Exercise 2.1 drills roster and set-builder conversion directly, Exercise 2.2 introduces special sets and the union/intersection/difference operations, Exercise 2.3 asks when two differently-written sets are actually the same set, and Exercise 2.4 closes the chapter by classifying sets as finite or infinite. If you'd like a refresher on the number systems these examples draw from, see Chapter 1's introduction.