A binary operation \(r\) is a function on a pair of sets \(A\) and \(B\). There exists a set \(C\) such that \[r : A \times B \rightarrow C\]
Given elements \(a \in A\) and \(b \in B,\) we write \(arb\) instead of \(r(a,b).\)
A binary operation on a set \(A\) is a function \[r : A \times A \rightarrow A\]
Addition is an operation on integers. \[+: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}\] Instead of writing \(+(2,3) = 5,\) we write \(2+3 = 5.\)
Scalar multiplication is an operation on \(\mathbb{R} \times M_{2 \times 2}\) where \(M_{2 \times 2}\) is the set of \(2 \times 2\) matrices.