A set is a well-defined collection of distinct objects. "Well-defined" means that, given any object, you can decide without ambiguity whether or not it belongs to the collection. The objects in a set are its elements (or members). We write $a \in A$ for "$a$ is an element of $A$" and $a \notin A$ for "$a$ is not an element of $A$".
"The collection of tall students" is not a set — "tall" is subjective. "The collection of students above 170 cm" is a set. Sets are named with capital letters; elements are usually written in lower case.
Two ways to represent a set:
In roster (tabular) form you list every element inside braces, separated by commas. Order does not matter and elements are never repeated — $\{1,2,2,3\}$ is just $\{1,2,3\}$. In set-builder form you state the property all elements share, read as "the set of all $x$ such that …". Use roster form when the elements are few or follow no neat rule (the days of the week); use set-builder when the elements are many, infinite, or share a clean property (all multiples of $7$).
Converting between the two forms is a core skill. To go roster $\to$ set-builder, hunt for the pattern: are the numbers squares, multiples, primes, terms of a sequence? To go set-builder $\to$ roster, read the property and the domain, then list the qualifying elements in order.
Standard number sets have reserved symbols you will use throughout the syllabus:
| Symbol | Set | Examples |
|---|---|---|
| $\mathbb{N}$ | Natural numbers | $1, 2, 3, \dots$ |
| $\mathbb{W}$ | Whole numbers | $0, 1, 2, 3, \dots$ |
| $\mathbb{Z}$ | Integers | $\dots, -2, -1, 0, 1, 2, \dots$ |
| $\mathbb{Q}$ | Rational numbers | $\tfrac{p}{q},\ q \ne 0$ |
| $\mathbb{R}$ | Real numbers | all points on the number line |
The empty set. The empty set $\varnothing$ (or $\{\,\}$) has no elements. It arises naturally whenever a defining condition can never be met — for example $\{x : x \in \mathbb{R},\ x^2 = -1\}$ is empty because no real number squares to a negative. Beware the common trap: $\{0\}$ and $\{\varnothing\}$ are not empty — the first holds the number $0$, the second holds one element (the empty set itself), so each has cardinality $1$.
Types of sets by size. A singleton has exactly one element. A set is finite if its elements can be counted and the counting comes to an end, and infinite otherwise. Crucially, "finite" is about the count terminating, not about being small: $\{x : x \in \mathbb{N},\ x < 10^{9}\}$ is finite, while the set of all primes is infinite. The number of elements in a finite set $A$ is its cardinal number, written $n(A)$; by definition $n(\varnothing) = 0$.
Equal sets. $A = B$ exactly when they have precisely the same elements — every element of $A$ is in $B$ and vice versa. Equivalent sets only need the same cardinal number: $\{a,b,c\}$ and $\{1,2,3\}$ are equivalent (both have $n = 3$) but not equal. Equal sets are always equivalent; equivalent sets need not be equal.
| Equal sets | Equivalent sets | |
|---|---|---|
| Condition | identical elements | same $n(A)$ only |
| Symbol | $A = B$ | $A \leftrightarrow B$ |
| Example | $\{1,2\} = \{2,1\}$ | $\{1,2\} \leftrightarrow \{a,b\}$ |
Deeper Insight — why "well-defined" and "distinct" carry the whole subject: Every idea in this chapter rests on the two words in the definition. Well-defined guarantees that membership is a yes/no question with no grey area, which is exactly what lets us reason about sets with certainty — and it is why a vague collection like "good books" can never be a set. Distinct is the reason $\{1,1,2\}=\{1,2\}$: a set records only which objects are present, never how many times or in what order. Confusing a set with a list (where order and repetition matter) is the single most common beginner error, and it quietly breaks later work — for instance, counting in Permutations and Combinations depends on knowing exactly when arrangement matters and when only membership does. Treat a set as a pure "yes/no membership table" and the operations that follow become almost obvious.