Set Theory

Basic Sets

N := Natural Number
Z := Integer
Q := Rational Number
R := Real Number
C := Complex Number

When two sets have no elements in common, they are said to be disjoint

$A^{'} = {x : x \in U and x \in / A}$

$A \ B = A \cap B^{'} = {x : x \in A and x \in / b}$

Subsets of $A \times B$ are called relations

A relation is well-defined if each element in the domain is assigned to a unique element in the range.

满射
each element in B has an A

单射
$a_{1} \neq = a_{2} \Leftrightarrow f (a_{1}) \neq = f (a_{2})$

Countable: can be put in 1-1 correspondence with positive integers
- Integers
- Rational Numbers
Uncountable
- Interval [0, 1]
- Read line, plane

双射
A map that is both one-to-one and onto is called bijective.

If S is any set, we will use ids or id to denote the identity mapping from S to itself.
Define this map by id(s)= s for all s $\in$ S.

A mapping is invertible if and only if it is both one-to-one and onto.