Sets and their representations

Membership must be decidable. A collection qualifies as a set only if, for every conceivable object, there is a definite yes-or-no answer to "does it belong here?" Sets are conventionally named with capital letters ($A, B, X, \dots$) and their elements with lowercase letters. If $x$ is an element of set $A$, this is written $x \in A$ (read "$x$ belongs to $A$"); if not, $x \notin A$.

Roster (tabular) form lists every element inside curly braces, separated by commas, each element written exactly once, with order not mattering:

where: $A$ is the set being named, and each entry between the braces is one element of $A$, written only once regardless of how many times it might naturally recur in the describing context.

Set-builder form instead states a shared property using a variable, so the reader can decide membership by checking the rule rather than scanning a list:

where: $x$ is the variable standing for a generic element, the colon (or equivalently a vertical bar) is read "such that," and the condition after it is the test every element must pass. Standard number-set symbols recur throughout the chapter: $\mathbb{N}$ (natural numbers), $\mathbb{Z}$ (integers), $\mathbb{Q}$ (rational numbers), $\mathbb{R}$ (real numbers) — always state which of these the variable is drawn from, since "$x \le 5$" alone is ambiguous without a domain. The cardinal number of a set $A$, written $n(A)$, is simply the count of its distinct elements. For the set above, $n(A) = 5$.

Roster form is convenient when a set is small or the list is obvious; set-builder form is essential once a set has infinitely many elements, or listing every one is impractical. The even natural numbers, for instance, have no last element to write down, so roster form cannot exhaust them, but set-builder form handles them compactly: $\{x : x \in \mathbb{N}, \; x \text{ is even}\}$. For a large but obviously patterned set, roster form can still be used with dots to signal the pattern continues, e.g. $\{1, 2, 3, \dots, 100\}$.

Both notations describe the same underlying object — a collection of distinct elements with no order — so switching between them never changes what the set actually contains; it only changes how conveniently that content can be written down or checked. Roster form is a direct inventory: to know whether something belongs, look at the list. Set-builder form is a test: to know whether something belongs, check the condition. This is why set-builder descriptions of a given set are rarely unique — $B = \{x : x = n^2, n \in \mathbb{N}, 1 \le n \le 5\}$ and $B = \{x : x \text{ is the square of a natural number}, x \le 25\}$ both pick out exactly $\{1, 4, 9, 16, 25\}$, because both rules admit precisely those five numbers and no others. What makes a rule valid set-builder notation is not its exact wording but that it draws the boundary around the same elements, with no ambiguity about which objects satisfy it. This is also why the "well-defined" requirement from the core idea is really the same requirement stated two ways: a collection is a legitimate set exactly when some rule — spoken, listed, or symbolic — can decide membership without doubt, whichever form is used to state it.

Write the set $A = \{x : x \text{ is an integer}, -3 \le x < 4\}$ in roster form, and find $n(A)$.

Step 1. Read the condition carefully: $x$ is an integer satisfying $-3 \le x < 4$, so the left endpoint $-3$ is included but the right endpoint $4$ is not.

Step 2. List every integer from $-3$ up to, but not including, $4$: that gives $-3, -2, -1, 0, 1, 2, 3$.

Step 3. Write the roster form using this list, each element once:

Step 4. Count the elements to get the cardinal number: there are 7 elements, so $n(A) = 7$.

Competitive stretch

Now go the other way: write $B = \{1, 4, 9, 16, 25\}$ in set-builder form. Step 1. Look for a common pattern among the elements: $1 = 1^2$, $4 = 2^2$, $9 = 3^2$, $16 = 4^2$, $25 = 5^2$, so each element is the square of a natural number from 1 to 5. Step 2. Express this as a rule with variable $x = n^2$: $B = \{x : x = n^2, \; n \in \mathbb{N}, \; 1 \le n \le 5\}$. Step 3. Check it is unambiguous — the domain restriction $1 \le n \le 5$ is essential, since without it the rule would describe an infinite set of squares instead of these five. The set-builder form is $B = \{x : x = n^2, n \in \mathbb{N}, 1 \le n \le 5\}$, with $n(B) = 5$.

• Writing a repeated element in roster form, e.g. $\{2, 2, 3\}$ — a set never repeats elements; the correct form is $\{2, 3\}$.
• Confusing $\in$ (relates an element to a set) with $\subseteq$ (relates two sets) — these connect different kinds of objects and cannot be swapped.
• Treating order as meaningful: $\{1, 2, 3\}$ and $\{3, 1, 2\}$ name the same set.
• Describing a collection with a vague, subjective property (e.g. "beautiful flowers") and calling it a set — the defining property must be objective and decidable for every object.
• Writing a set-builder rule without stating the domain of the variable — "$x \le 5$" alone is ambiguous; specify $x \in \mathbb{N}$, $x \in \mathbb{Z}$, or whichever domain is intended.

Ask the tutor: Not sure whether a tricky collection is a well-defined set, or want more practice converting between roster and set-builder form — ask the AI tutor to work through extra examples with you.