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).

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