If is a set, the collection of all its subsets also forms a set, denoted as or , called the power set.
The Cartesian product between and is the set consisting of all ordered pairs of elements in and elements in . We can write this as
Functions
A function or a map between and is a relation between and , such that for any , there is a unique , such that . We denote this as . We often write such a function as
is the domain, is the codomain.
Relation of Functions
function is called an injection if implies .
It is called a surjection if the range equals codomain. . Or, for any , such that .
It is called a bijection if it is both an injection and a surjection.
identity function
The identity function from A to itself is defined as . AKA,
If , , , we call the inverse of ,denoted as , and to be invertible.
Group
Definition
A group is a pair , where is a set, a map, such that:
function: is well-defined
identity: There is an element , such that for any , .
associative: for any , .
inverse: for any , there is some other element such that . Later will be shown to be unique, and we denote it as .
can also be written as or or . The identity can also be denoted as or . When is a group we can also say “ is a group under ”.
Note: To show a function is well-defined is also proving that the closure of the group. For every pair of elements in the group, the result of the group operation (often denoted as or ) on and must also be in the group. i.e. on . For any we have since , . So, is well-defined.
Note. Well-Defined means
for some such that implies .
one-to-one of implies .
Abelian Group
If is commutative, i.e. for any , , then we call it a commutative group or abelian group.
. This is called the trivial group, often denoted as or .
Example
For there is a map action (composition action) . Show is abelian group.
\begin{proof}
Suppose , are two elements in group, then for any ,
Hence is abelian. \end{proof}
Permutation Group
is any non-empty set, (bijections from to , ). This is called the permutation group or symmetric group. When is a finite set of elements this is denoted as . are all finite, and is abelian iff .
Example
Homomorphisms
Definition
If and are two groups, satisfies that for any , , then we call a homomorphism. If is bijective then we call an isomorphism.