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 . This actually hold true for all integers (positives and negatives).

But clearly you would agree that subtraction is not associative. .

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

It follows that the additive inverse of is as . Therefore we chose to write the additive inverse of as .

Let us now have a look at which we write so that clearly subtraction is not associative

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