Set notation is a systematic method for describing collections of objects, typically numbers, using curly brackets, denoted as $\{\}$.

In set notation, the elements of a set are enclosed within curly brackets and separated by commas. For instance, the set that includes the numbers $1$, $2$, and $3$ is represented as $\{1, 2, 3\}$. When a set can be defined by a specific pattern or rule, we can use set-builder notation. For example, the set of all even numbers can be expressed as

which translates to “the set of all $x$ such that $x$ is an even number.”

Set notation also employs various special symbols. The symbol $\in$ signifies “is an element of.” Thus, if we define a set $A = \{1, 2, 3\}$, we can express that $2$ is an element of $A$ by writing $2 \in A$. Conversely, the symbol $\notin$ indicates “is not an element of.” Therefore, $4 \notin A$ denotes that $4$ is not an element of $A$.

Sets can also be characterized by their properties. For example, the set of all positive integers that are less than $10$ can be represented as

where $\mathbb{N}$ denotes the set of natural numbers. This reads as “the set of all $x$ in natural numbers such that $x$ is less than $10$.”

Additionally, there are several special sets that have unique symbols. The empty set, which contains no elements, is denoted by either $\emptyset$ or $\{\}$. The set of all natural numbers is represented by $\mathbb{N}$, the set of all integers by $\mathbb{Z}$, the set of all rational numbers by $\mathbb{Q}$, and the set of all real numbers by $\mathbb{R}$.

Understanding set notation is essential in mathematics as it provides a clear and concise framework for describing and manipulating collections of objects.