Binary operation is the starting point of group theory. Before we can define a group, we must first know how two elements of a set are combined and whether the result remains inside the same set. This is the first structural idea in abstract algebra, because the set alone does not determine the algebraic system. The same set may behave very differently under addition, multiplication, subtraction, composition, or a specially defined rule. In this lecture, students will learn the meaning of a binary operation, the closure requirement, and the method for checking whether a given rule is a binary operation on a given set.

Let $S$ be a non-empty set. A rule $*$ is called a $\textbf{binary operation}$ on $S$ if $*$ assigns to every ordered pair $(a,b)\in S\times S$ a unique element $a*b\in S$. Equivalently, $*$ is a function $$\begin{equation*} *:S\times S\to S. \end{equation*}$$ Thus a binary operation on $S$ must satisfy the following conditions: (i) $a*b$ is defined for every $a,b\in S$. (ii) $a*b\in S$ for every $a,b\in S$.