BASIC MATH NOTATION AND DEFINITIONS

Set

N, Z, Q, R, C.

• natural numbers
• integers
• rationals
• reals
• complex numbers

Operations

power set

• power set $2^X$

  all subsets

Function

$f: X\rightarrow Y$

• graph 图像

  有序x和y对组成的集合

• image 像

  当元素x 可映射至元素y，则y是x的“像”，而x为y的“原像”（inverse image, pullback)

• identity function $id_x: X\rightarrow X$

性质

• injective 单射

  left inverse 存在左逆，为满射

• surjective 满射

  right inverse 存在右逆，为单射

• bijective 双射

  既是单射又是漫射

Operation

Composition

Inverse

Binary Relation

$xRy\Leftrightarrow (x,y)\in R$

性质

• transitive
• reflexive
• symmetric
• antireflexive
• antisymmetric

运算

Composition

$R\circ S$

乘幂：

• $R^0=I_X$
• $R^{n+1}=R\circ R^n$

converse

$R^{-1}={(x,y)\in Y\times X|(y,x)\in R}$

closure

• transitive closure: is the smallest transitive relation containing R

  $R^+=\bigcup\limits_{n\ge1} R^n$

• reflexive and transitive closure: is the smallest transitive and reflexive relation containing R

  $R^*=\bigcup\limits_{n\ge0}R^n=R^+\cup I_X$

重要关系

• 等价关系
• 偏序关系