For any natural numbers $a,b,c$, prove the associativity property $(a + b) + c = a + (b + c)$.

Question

For any natural numbers $a,b,c$, we have $(a + b) + c = a + (b + c)$.

MY ATTEMPT

We shall prove it by induction on $c$. For $c = 0$, we have that $(a + b) + 0 = a + b$ and $a + (b + 0) = a + b$. Let us assume that $(a + b) + c = a + (b + c)$ for a natural number $c$ and prove the relation holds for $c\texttt{++}$. Indeed, one has \begin{align*} (a + b) + c\texttt{+}\texttt{+} = ((a + b) + c)\texttt{+}\texttt{+} = (a + (b + c))\texttt{+}\texttt{+} = a + (b + c)\texttt{+}\texttt{+} = a + (b + c\texttt{+}\texttt{+}) \end{align*}

Can someone check if I am reasoning rightly?

Answer 1

Proof by induction :

$\displaystyle (a + b) + c^+$ $\hspace{0.25in}$

$=\displaystyle((a + b) + c)^+$$\hspace{0.195in}$Definition of Addition in Minimal Infinite Successor Set

$=\displaystyle (a + (b + c))^+$$\hspace{0.195in}$Induction Hypothesis

$=\displaystyle a + ((b + c)^+)$$\hspace{0.25in}$Definition of Addition in Minimal Infinite Successor Set

$=\displaystyle a + (b + c^+)$$\hspace{0.32in}$Definition of Addition in Minimal Infinite Successor Set

So $\ P (c) \implies P (c^+)$ and the result follows by the Principle of Mathematical Induction.