# Definition:Subtraction/Integers

## Definition

The **subtraction** operation in the domain of integers $\Z$ is written "$-$".

As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same.

Thus subtraction can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers.

That is, the integers being defined as all the difference congruence classes, integer multiplication can be defined directly as the operation induced by natural number multiplication on these congruence classes.

It follows that:

- $\forall a, b, c, d \in \N: \eqclass {\tuple {a, b} } \boxminus - \eqclass {\tuple {c, d} } \boxminus = \eqclass {\tuple {a, b} } \boxminus + \tuple {-\eqclass {\tuple {c, d} } \boxminus} = \eqclass {\tuple {a, b} } \boxminus + \eqclass {\tuple {d, c} } \boxminus$

Thus **integer subtraction** is defined between all pairs of integers, such that:

- $\forall x, y \in \Z: x - y = x + \paren {-y}$

## Also known as

In the context of mathematical logic it is sometimes referred to as **proper subtraction** so as to distinguish it from the partial subtraction operation as defined on the natural numbers.

## Sources

- 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*... (previous) ... (next): $1$: Complex Numbers: The Real Number System: $2$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 20$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.1$: The integers