 We begin with the concept of a binary operation on a set \(X\), namely a function \(f: X^2 \rightarrow X\) such that \(f(x, y)\) is an element of \(X\) for all elements \(x, y\) of \(X\). Such an operation is said to be associative when it satisfies \(f(f(x, y), z) = f(x, f(y, z))\) for all \(x, y, z\) in \(X\).