8. Introduction to Algebraic Structures¶
- Group: a group G denoted by
is a set G together with a binary operation on G such that the following CAIN axioms hold:- Closure:
- Associativity:
- Identity:
- Inverse:
- Closure:
- When the CAIN properties are true for a set and operation on it; we call it a group.
- Abelian Group: a group G is called an abelian group if the previous CAIN axioms hold, and the following Commutativity axiom holds:
- Commutativity:
- Commutativity:
-
List of common sets:
: Natural numbers (integers greater than 0) : Integers : Negative integers : Positive integers (excluding 0) : Rational numbers : Real numbers : Complex numbers : Integers modulo n : Positive integers : Integers modulo n excluding 0 : Whole numbers
-
Semigroup: a semigroup S denoted by
is a set S together with a binary operation on S such that the following CA axioms hold:- Closure:
- Associativity:
- Closure:
-
Monoid: a monoid M denoted by
is a set M together with a binary operation on M such that the following CAI axioms hold (that is, a semigroup with identity):- Closure:
- Associativity:
- Identity:
- Closure:
Property | Meaning | English * (approximation, not very accurate) |
---|---|---|
Closure | All elements and results of applying |
|
Associativity | Parentheses do not change order | |
Identity | There is an element that does not change the results of all other elements | |
Inverse | For every element there is an element that cancel it out (according to the |
|
Commutativity | Order does not change the result of |
Name | Closure | Associativity | Identity | Inverse | Commutativity | Example |
---|---|---|---|---|---|---|
Semigroup | Yes | Yes | ||||
Monoid | Yes | Yes | Yes | |||
Group | Yes | Yes | Yes | Yes | ||
Albelian Group | Yes | Yes | Yes | Yes | Yes |
- Examples:
Set and Operation | Closure | Associativity | Identity | Inverse | Commutativity | Example |
---|---|---|---|---|---|---|
Matrix Addition 3 * 3 Real matrices | Yes | Yes | Yes (0 matrix) | |||
Matrix multiplication 2 * 2 of even integers | Yes | Yes | No (1 is not in set) | |||
Addition on Set of integers | Yes | Yes | Yes (0) | Yes | Yes | |
Multiplication on Set of odd integers | Yes | Yes | Yes (1) | Yes | Yes |
Modular Arithmetic¶
- Congruence (mod n):
if - Example:
- because
and and . - Think of it as a clock, the 15hour and 3hour are the same pointing to number 3.
- Available classes for
are or where
- because
- Available classes for
are - Congruence Class (mod n):
- Example:
because - The common divisor of a and b is a number that divides both a and b.
- The greatest common divisor of a and b is the largest number that divides both a and b.
- Two numbers are relatively prime if their greatest common divisor is 1.
References¶
- Neso Academy. Group and Abelian Group. https://www.youtube.com/watch?v=8TjYHK804mU