Associative law with negative sign

Question

We know that $a+b+c$ is meaningful for, say, the group $\left( \mathbb{Z}% ,+\right) $. Since for any $b,c$, we must have $b^{-1},c^{-1}$, therefore $% a+b^{-1}+c^{-1}$ has to be meaningful, too, but we know that $a-b-c$ is meaningless because the associative law fails. What's going on?

Answer 1

The expression $a - b - c$ is not meaningless, it just needs to be interpreted in the correct way. As you rightly point out, subtraction is not associative, so we need to specify whether $a - b - c$ means $(a - b) - c$ or $a - (b - c)$. As you are probably aware, the expression $a - b - c$ refers to the former.

The fact that subtraction and division are not associative, is precisely why we have PEMDAS/BEDMAS/other variants which describe in which order to perform operations; note, these rules tell you that $a - b - c$ means $(a - b) - c$ and not $a - (b - c)$. This allows us to use expressions with many binary operations and no parentheses, yet still have a well-defined meaning.