Algebra and Number Theory/Printable version
This is the print version of Algebra and Number Theory You won't see this message or any elements not part of the book's content when you print or preview this page. |
The current, editable version of this book is available in Wikibooks, the open-content textbooks collection, at
https://en.wikibooks.org/wiki/Algebra_and_Number_Theory
Sets
![]() | This page may fit the criteria for speedy deletion for this reason:
abandoned, very little meaningful content Please share your thoughts.
Do you think this page should be kept or doesn't fit the criteria? Want to discuss this with more people? You can continue to edit this page, which may save it from deletion if improved. Administrators: Please check the page history, page log, and especially the last edit, before deleting. |
Set Operations
[edit | edit source]Special Sets
[edit | edit source]Functions and Binary Operations
[edit | edit source]Equivalence Relations
[edit | edit source]
Elementary Number Theory
Divisibility
[edit | edit source]Definition 1: (divides, divisor, multiple)
Let , with . We say that " divides " or that " is a multiple of ", if there exists some such that .
We write this as .
Proposition 1: (some elementary properties of division)
Let be integers. Then
- If and , then . ▶ □
- If and , then .
- If and , then .
- If and , then . ▶ □
Examples: because . However : if it did, would also divide (by Proposition 1, point 3), which is impossible (Proposition 1, point 1). Similarly, .
Proposition 2: (division algorithm)
Let , with . Then , for some , with .
▶