$A$ is a subset of $B$, written $A \subseteq B$, if every element of $A$ is also an element of $B$. Formally:
If $A \subseteq B$ but $A \ne B$ (so $B$ has at least one element not in $A$), then $A$ is a proper subset, written $A \subset B$. Two facts hold for every set $A$: the empty set is a subset of it ($\varnothing \subseteq A$), and every set is a subset of itself ($A \subseteq A$). A key consequence is the test for equality — $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$ — which is the standard way to prove two sets are the same.
Element vs subset — do not confuse $\in$ and $\subseteq$. The symbol $\in$ relates an object to a set ($2 \in \{1,2,3\}$); $\subseteq$ relates a set to a set ($\{2\} \subseteq \{1,2,3\}$). Writing $2 \subseteq \{1,2,3\}$ or $\{2\} \in \{1,2,3\}$ is wrong.
Subsets of the real numbers. The number systems are nested as subsets: $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$. The irrationals $\mathbb{R} \setminus \mathbb{Q}$ (such as $\sqrt{2}$ and $\pi$) are also a subset of $\mathbb{R}$.
Universal set $U$ is the fixed "background" set containing all objects under discussion for a particular problem. Every other set in that problem is a subset of $U$. The choice of $U$ depends on context — it may be $\mathbb{R}$ in a number problem, or "all students of a class" in a survey problem.
Intervals are subsets of $\mathbb{R}$ describing continuous stretches of the number line. A square bracket includes the endpoint; a round bracket excludes it. The symbols $\infty$ and $-\infty$ are not numbers, so the bracket beside them is always round:
| Interval | Set-builder | Endpoints |
|---|---|---|
| $[a, b]$ | $\{x : a \le x \le b\}$ | both included (closed) |
| $(a, b)$ | $\{x : a < x < b\}$ | both excluded (open) |
| $[a, b)$ | $\{x : a \le x < b\}$ | left included only (half-open) |
| $(a, b]$ | $\{x : a < x \le b\}$ | right included only (half-open) |
| $(a, \infty)$ | $\{x : x > a\}$ | $\infty$ is never included |
| $(-\infty, b]$ | $\{x : x \le b\}$ | $-\infty$ is never included |
The length of a bounded interval with endpoints $a$ and $b$ is $b - a$, regardless of which endpoints are included.
Power set: the set of all subsets of $A$, written $P(A)$. Its elements are themselves sets — note that $\varnothing \in P(A)$ and $A \in P(A)$ for every $A$. The count of subsets is fixed by the size of $A$:
The reason is a counting argument: to build a subset you make one independent yes/no choice per element — "include it or not" — giving $2 \times 2 \times \dots \times 2 = 2^m$ possibilities. Of these, $2^m - 1$ are proper subsets (all except $A$ itself), and $2^m - 2$ are proper non-empty subsets (excluding $A$ and $\varnothing$). The number of subsets containing one fixed element is $2^{m-1}$ — exactly half — because the choice for that element is settled and the remaining $m-1$ elements are still free.
Deeper Insight — subset is the backbone, $2^m$ is its first dividend: The subset relation $\subseteq$ is the most important relation in set theory because nearly every later definition is phrased through it — equality ($A=B$ means $A\subseteq B$ and $B\subseteq A$), the complement, even the way a universal set frames a problem. Keep two edge cases permanently in mind: $\varnothing$ is a subset of everything (vacuously — there is no element of $\varnothing$ that could fail the test), and every set is a subset of itself. The power-set formula $2^m$ is your first real glimpse of why these ideas matter beyond bookkeeping: it is the same "independent binary choice" logic that powers the Binomial Theorem ($\sum \binom{m}{k} = 2^m$, the total number of subsets counted by size) and probability over finite outcomes. Once you see a subset as a string of yes/no decisions, $2^m$ stops being a formula to memorise and becomes something you can re-derive on the spot.