Set¶

Definition¶

A set $A$ is an unordered collection of objects. If $x$ is an object, we say that $x$ is an element of $A$ or write $x\in A$ if $x$ lies in the collection; otherwise, we write $x\notin A\text{.}$

Interpretation¶

A set collects objects without imposing order: all that matters is whether $x\in A$ or $x\notin A$$A$. Order and repetition play no role, so $\{1,2,3\}$, $\{3,1,2\}$ and $\{1,1,2,3\}$ are the same set.

A set can be given by listing its elements, as in $\{0,1,2,3\}$, or by a condition, as in $\{n\in\mathbb{N}\colon n^2<10\}$$10\}$, read "the set of natural numbers $n$ such that $n^2<10$$10$"; these two descriptions give the same set. Two sets are equal when they have the same elements, so equality is usually shown by proving both inclusions, $A\subseteq B$ and $B\subseteq A$$A$. Elements may themselves be sets: $\{\varnothing\}$ has one element, the empty set.

A common slip is to confuse "is an element of" with "is a subset of": $1\in\{1,2\}$ and $\{1\}\subseteq\{1,2\}$$\{1,2\}$, but $\{1\}\notin\{1,2\}$$\{1,2\}$, because the elements of $\{1,2\}$ are the numbers $1$ and $2$, not sets.

Ignoring order is what separates a set from a sequence, a list in which order and repetition matter.

Key Results¶

• Sets are defined by membership, not order.
• Equality of sets depends on having the same elements.
• Sets may be finite or infinite.

Examples¶

• $\{1,\,2,\,3\}$ is a set.
• We have $\pi\notin \mathbb{N}\text{.}$

Related Concepts¶

• Sequence
• Solution Set
• Axioms of Probability

Sources¶

• Math I: Calculus · Lecture 1

Symbols: element of, natural numbers, pi, subset

Last updated: 20 October 2026 03:02