Integers under Subtraction form Magma

Theorem

The set of integers $\Z$ under the operation of subtraction forms a magma.

Proof

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 subtraction is defined as:

$\forall a, b \in \Z: a - b := a + \paren {-b}$

Recall that the Integers under Addition form Group.

Hence:

$\forall b \in \Z: -b \in \Z$

and:

$\forall a, b \in \Z: a + \paren {-b} \in \Z$

That is:

$\forall a, b \in \z: a - b \in \Z$

Hence the result by definition of magma.

$\blacksquare$

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