Integers - kipawaa/Proof-Tree GitHub Wiki

Statement

Common Definition

The integers are a set of numbers $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$.

Explanation

This set is an extension of the Natural Numbers to include negative numbers.

Proof(s)

Set-Theoretic Construction

The integers are given by the Equivalence Relation $(a, b) \sim (c, d)$ iff $a + d = b + c$ for $(a, b), (c, d) \in \mathbb{Z} \times \mathbb{Z}$.

History

Applications

Links

Dependencies

• Natural Numbers
• Equivalence Relation

Dependents

• Integers are Countable
• Rational Numbers

Sources

• Hrbacek, K., & Jech, T. (1999). Introduction to Set Theory, Revised and Expanded (3rd ed.). CRC Press. https://doi.org/10.1201/9781315274096