Element of a Set
An element is an individual object that belongs to a set, written using the membership symbol ∈.
Formula
x \in A \text{ (x is an element of A)}
Definition
An element is simply one of the things inside a set; the symbol $\in$ means "is a member of" and $\notin$ means "is not a member of." For a set $A$, $x \in A$ means $x$ is an element of $A$, and sets can contain any type of object, including other sets, so $\{\{1,2\}, \{3,4\}\}$ has two elements, each of which is itself a set. In ZFC, the membership relation $\in$ is the primitive undefined notion from which all set theory is built, and every element is itself a set; the iterative hierarchy $V_0 = \emptyset$, $V_{\alpha+1} = P(V_\alpha)$, $V = \bigcup V_\alpha$, is the universe of ZFC sets, built inductively from membership.
Example
In the set $A = \{2, 4, 6, 8\}$: $4 \in A$, but $5 \notin A$. Let $A = \{1, \{2, 3\}, 4\}$: then $1 \in A$, $\{2,3\} \in A$, and $4 \in A$, but $2 \notin A$ since $2$ is inside a nested set, not directly in $A$. The empty set $\emptyset$ is an element of $\{\emptyset\}$, and $\emptyset \subseteq$ every set, so $\emptyset$ and $\{\emptyset\}$ are different: $\emptyset$ has $0$ elements, $\{\emptyset\}$ has $1$ element.
Key Insight
Think of a set as a club and its elements as members: the symbol $\in$ asks, "Is this person in the club?" The distinction between an element and a subset trips many students: $\{2\} \subseteq A$ because $\{2\}$ is a subset, but $2 \in A$ because $2$ (not $\{2\}$) is the element directly in $A$. In type theory, an alternative foundation, elements have types that restrict membership, avoiding paradoxes differently than ZFC.