Subtraction on Numbers is Not Associative

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Theorem

The operation of subtraction on the numbers is not associative.

That is, in general:

$a - \paren {b - c} \ne \paren {a - b} - c$


Proof

By definition of subtraction:

\(\displaystyle a - \paren {b - c}\) \(=\) \(\displaystyle a + \paren {-\paren {b + \paren {-c} } }\)
\(\displaystyle \) \(=\) \(\displaystyle a + \paren {-b} + c\)


\(\displaystyle \paren {a - b} - c\) \(=\) \(\displaystyle \paren {a + \paren {-b} } + \paren {-c}\)
\(\displaystyle \) \(=\) \(\displaystyle a + \paren {-b} + \paren {-c}\)

So we see that:

$a - \paren {b - c} = \paren {a - b} - c \iff c = 0$

and so in general:

$a - \paren {b - c} \ne \paren {a - b} - c$

$\blacksquare$


Sources