Subset not necessarily Submagma

Theorem

Let $\struct {S, \circ}$ be a magma.

Let $T \subseteq S$.

Then it is not necessarily the case that:

$\struct {T, \circ} \subseteq \struct {S, \circ}$

That is, it does not always follow that $\struct {T, \circ}$ is a submagma of $\struct {S, \circ}$.

Let $\struct {\Z, -}$ be the magma which is the set of integers under the operation of subtraction.

We have that the natural numbers $\N$ are a subset of the integers.

Consider $\struct {\N, -}$, the natural numbers under subtraction.

For example:

$1 - 2 = -1 \notin \N$

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

So $\struct {\N, -}$ is not a submagma of $\struct {\Z, -}$

Hence it is not true to write $\struct {\N, -} \subseteq \struct {\Z, -}$, despite the fact that $\N \subseteq \Z$.

Thus $\struct {\N, -}$ is not a submagma of $\struct {\Z, -}$.

$\blacksquare$