## Associativity is not always given

Many of my students tends to believe that associativity of a given operation is always true.
We have seen in the previous post that addition is associative so that we always have $(a+b)+c=a+(b+c)$. This actually hold true for all integers (positives and negatives).

But clearly you would agree that subtraction is not associative. $(a-b)-c\ne a-(b-c)$.

This has obviously to do with the way subtraction is define. One way to think of it is to write $-a$ as the additive inverse (or opposite) of $a$, such that: $a+(-a)=0$. The subtraction $b-a$ is then defined as $b+(-a)$. The rule is that we can replace $+(-a)$ directly by $-a$.

It follows that the additive inverse of $b+c$ is $(-b)+(-c)$ as $b+c+(-b)+(-c)=0$. Therefore we chose to write the additive inverse of $b+c$ as $-(b+c)$.

Let us now have a look at $(a-b)-c=(a+(-b))+(-c)=a+((-b)+(-c))=a+(-(b+c))$ which we write $a-(b+c)$ so that clearly subtraction is not associative

$(a-b)-c=a-(b+c)$.