Sets, subsets and intervals

Describing a set. A set can be written in two equivalent ways. Roster (tabular) form simply lists the elements inside braces, e.g. $A = \{2, 4, 6, 8\}$. Set-builder form states a rule every element must satisfy, e.g. $A = \{x : x \text{ is an even natural number}, x \le 8\}$. The two describe the same set; roster form is convenient for small, explicit sets, set-builder form for sets defined by a condition (especially infinite ones, which cannot be listed at all).

Special kinds of sets. The empty set $\varnothing$ (also written $\{\}$) has no elements at all — it is not the same as $\{0\}$, which contains one element, the number $0$. A set is finite if the process of counting its elements terminates, and infinite otherwise; $\mathbb{N}$, the natural numbers, is the standard example of an infinite set. Two sets $A$ and $B$ are equal, written $A = B$, exactly when they contain the same elements — order and repeated listing never matter, so $\{1,2,2,3\} = \{1,2,3\}$.

Subsets and the power set. $A$ is a subset of $B$, written $A \subseteq B$, if every element of $A$ also belongs to $B$:

where: $x \in A$ reads "$x$ is a member of $A$", and $\implies$ is logical implication — the definition says every membership test that passes for $A$ also passes for $B$. If in addition $A \neq B$, then $A$ is a proper subset, written $A \subset B$: $B$ has at least one element $A$ lacks. Every set satisfies $A \subseteq A$ (reflexivity), and $\varnothing \subseteq A$ for every set $A$, vacuously, since $\varnothing$ has no element that could fail the test. The power set $P(A)$ collects all the subsets of $A$, proper or not, together with $A$ itself and $\varnothing$; a set with $n$ elements has a power set of size $2^n$ (proved below). A universal set $U$ is the fixed background set — everything under discussion in a given problem — relative to which subsets and complements are defined.

Intervals. An interval is a subset of $\mathbb{R}$ made of every real number lying between two fixed endpoints $a$ and $b$. Four flavours cover every case, depending on whether each endpoint is included:

where: a round bracket excludes the endpoint next to it (strict inequality) and a square bracket includes it (inequality with equality allowed); the same logic gives the two mixed cases $(a,b] = \{x : a < x \le b\}$ and $[a,b) = \{x : a \le x < b\}$. On a number line, an excluded endpoint is drawn as a hollow (open) circle and an included one as a filled (closed) circle, with the segment between them shaded.

Problem. Let $A = \{x \in \mathbb{N} : x^2 < 30\}$. (a) Write $A$ in roster form and state whether it is finite. (b) Find $n(P(A))$. (c) A set $B$ satisfies $-3 \le x < 5$ for $x \in \mathbb{R}$; write $B$ in interval notation.

Step 1. Find every natural number $x$ with $x^2 < 30$: testing $x = 1,2,3,4,5,6,\dots$ gives $1^2=1,\ 2^2=4,\ 3^2=9,\ 4^2=16,\ 5^2=25 < 30$, but $6^2=36 \not< 30$, so the search stops at $5$.

Step 2. Collect the results in roster form: $A = \{1, 2, 3, 4, 5\}$. Counting its elements terminates, so $A$ is finite, with $n(A) = 5$.

Step 3. Apply the power-set count from the derivation above, with $n = 5$:

Step 4. For part (c), the condition $-3 \le x < 5$ includes the left endpoint $-3$ (square bracket) and excludes the right endpoint $5$ (round bracket), giving $B = [-3, 5)$ — a filled dot at $-3$ and a hollow circle at $5$ on the number line, exactly the picture in the figure above with $a=-3,\ b=5$.

Final answers: $A = \{1,2,3,4,5\}$ is finite with $n(P(A)) = 32$, and $B = [-3, 5)$.

• Writing $\{0\}$ when the empty set is meant. $\{0\}$ contains one element, the number $0$; it is not empty. The empty set is only $\varnothing$ or $\{\}$, never $\{0\}$ or $\{\varnothing\}$.
• Confusing $\subseteq$ (subset, allows $A = B$) with $\subset$ (proper subset, forbids $A = B$). Every set satisfies $A \subseteq A$, but $A \subset A$ is always false.
• Swapping round and square brackets in interval notation. $(a,b)$ excludes both endpoints while $[a,b]$ includes both; one misplaced bracket describes an entirely different set.
• Undercounting the power set. $n(P(A)) = 2^n$ counts every subset, including $\varnothing$ and $A$ itself — students often list only the "obviously smaller" subsets and forget these two.

Ask the tutor: Stuck telling apart $\subseteq$ and $\subset$, or converting an inequality into interval notation? Ask the AI tutor to walk through a few practice sets with you.