# Category:Integer Addition

This category contains results about **Integer Addition**.

The **addition operation** in the domain of integers $\Z$ is written $+$.

We have that the set of integers is the Inverse Completion of Natural Numbers.

Thus 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 addition 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 addition can be defined directly as the operation induced by natural number addition on these congruence classes:

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

## Subcategories

This category has the following 4 subcategories, out of 4 total.

### A

### I

## Pages in category "Integer Addition"

The following 22 pages are in this category, out of 22 total.

### I

- Integer Addition Identity is Zero
- Integer Addition is Associative
- Integer Addition is Cancellable
- Integer Addition is Closed
- Integer Addition is Commutative
- Integer Addition is Well-Defined
- Integer Multiples Closed under Addition
- Integer Multiples Greater than Positive Integer Closed under Addition
- Integer Multiplication Distributes over Addition
- Integers under Addition form Abelian Group
- Integers under Addition form Infinite Cyclic Group
- Integers under Addition form Monoid
- Integers under Addition form Semigroup
- Integers under Addition form Totally Ordered Group
- Inverse for Integer Addition