Subtraction on Numbers is Anticommutative/Natural Numbers
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Theorem
The operation of subtraction on the natural numbers $\N$ is anticommutative, and defined only when $a = b$:
That is:
- $a - b = b - a \iff a = b$
Proof
$a - b$ is defined on $\N$ only if $a \ge b$.
If $a > b$, then although $a - b$ is defined, $b - a$ is not.
So for $a - b = b - a$ it is necessary for both to be defined.
This happens only when $a = b$.
Hence the result.