Natural Numbers under Subtraction do not form Magma

Theorem

The set of natural numbers $\N$ under the operation of subtraction does not form a magma.

Proof

Proof by Counterexample:

Recall the definition of magma:

$\struct {S, \circ}$ is a magma if and only if $\forall a, b \in S: a \circ b \in S$:

That is, a magma is closed under its operation.

Recall that the operation of natural number subtraction of $n - m$ is defined as:

$n - m = p$

where $p \in \N$ such that $n = m + p$.

But this is defined only when $n \ge m$.

Consider $4 - 7$.

Then there exists no $p \in N$ such that $4 = 7 + p$.

Thus $4 - 7$ is not defined on $\N$.

Thus the algebraic structure $\struct {\N, -}$ is not closed.

Hence the algebraic structure $\struct {\N, -}$ is not a magma by definition.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): groupoid
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): groupoid