# For any natural numbers $a,b,c$,prove that associativity of the product $(a\times b)\times c = a\times(b\times c)$.

For any natural numbers $$a,b,c$$, we have that $$(a\times b)\times c = a\times(b\times c)$$.

MY ATTEMPT

We shall prove it by induction on $$c$$. For $$c = 0$$, one has that \begin{align*} (a\times b)\times 0 = 0 = a\times 0 = a\times(b\times 0) \end{align*}

Then we are going to assume that $$(a\times b)\times c = a\times(b\times c)$$ and prove the proposed relation holds for $$c\texttt{+}\texttt{+}$$. Indeed, one has that \begin{align*} (a\times b)\times(c\texttt{+}\texttt{+}) & = (a\times b)\times c + a\times b = a\times(b\times c) + a\times b\\\\ & = a\times((b\times c) + b) = a\times(b\times(c\texttt{+}\texttt{+})) \end{align*} and we are done.

• For natural numbers I would write $a\cdot b$ and not $a\times b$ because this usually is the cross product which need not be associative, see here. – Dietrich Burde Mar 5 at 19:54