# 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 tuples 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.

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

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 20$ - 1974: Murray R. Spiegel:
*Theory and Problems of Advanced Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $2$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $2$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.1$: Chapter $2$: Integers and natural numbers: The integers - 1994: H.E. Rose:
*A Course in Number Theory*(2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization