> [!NOTE] Definition > Let $S$ be a set. A *binary operation* $f$ on $S$ is any [[Function|mapping]] $f: S\times S \to S$ # Properties We say $f$ is [[Commutativity|commutative]] iff $f(x,y)=f(y,x)$. We say $f$ is [[Associativity|associative]] iff $f(x,f(y,z))=f(f(x,y),z)$. We say that $e\in S$ is the [[Identity element of a binary operation|identity]] of $f$ if $f(e,x)=f(x,e)=x$ for all $x\in S$. Take $a,b\in S$. Then we say that $a$ is the [[Inverse under a binary operation|inverse]] of $b$ under $f$ iff $f(a,b)=e$. # Applications **Algebra**: (Group) We call the [[Algebraic Structure|algebraic structure]] $(S,f)$ a [[Groups|group]] if $f$ is associative, the identity of $f$ lies in $S$ and every member of $S$ has an inverse in $S$ under $f.$