Millan's Math

This is a proof of the associative property of natural number addition. We will show that for any natural numbers $$ a, b, and\, c $$ , we have $$ (a+b)+c = a+(b+c) $$ . To show this, let's induct on $$ c $$ . First, let's deal with the base case where $$ c=0$$ : $$ c=0 $$ Assumed $$ b+0=b $$ Lemma 2.2.2 $$ (a+b)+c=(a+b)+0 $$ Substitution $$ =a+b $$ Lemma 2.2.2 $$ =a+(b+0) $$ Substitution Therefore, $$ (a+b)+0=a+(b+0) $$ , proving the base case. Now, the inductive case. Here we are assuming that $$ (a+b)+c=a+(b+c)$$ for some natural number $$ c $$ . $$ (a+b)+(c++)=(c++)+(a+b) $$ Commutative property $$ =(c+(a+b))++ $$ Definition of addition $$ =((a+b)+c)++ $$ Commutative property $$ =(a+(b+c))++ $$ Inductive hypothesis $$ =((b+c)+a)++ $$ Commutative property $$ =((b+c)++)+a $$ Definition of addition $$ =((c+b)++)+a $$ Commutative property $$ =((c++)+b)+a $$ Definition of a