# Category:Subtraction

This category contains results about **Subtraction**.

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:

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

## Subcategories

This category has the following 8 subcategories, out of 8 total.

## Pages in category "Subtraction"

The following 16 pages are in this category, out of 16 total.

### I

### S

- Subtraction has no Identity Element
- Subtraction of Fractions
- Subtraction of Multiples of Divisors obeys Distributive Law
- Subtraction on Integers is Extension of Natural Numbers
- Subtraction on Numbers is Anticommutative
- Subtraction on Numbers is Not Associative
- Subtraction/Examples
- Subtraction/Examples/x+3 = 5