群 (Group)
Def: 一集合 G，與二元運算子(operator) * (稱之為乘法)，滿足：
    1. 封閉律：對任意運算元(operand) a, b ∈ G (∈ or [-代表屬於的意思)，則 a*b ∈ G
    2. 結合律：對任意 a, b, c ∈ G，則 (a*b)*c = a*(b*c)
    3. 存在單位元素：存在 e ∈ G，使得對任意 g ∈  G，e*g = g*e = g
    4. 存在反元素：對任意 g ∈ G，存在 h ∈  G，使得 g*h = h*g = e
    符合以上四點之 (G,*)，稱之為 "群"
    
交換群 (commutative group, abelian group)
Def: 當一群 (G,*)，滿足：
    對任意 a, b ∈ G，a*b = b*a 時，
    (G,*) 稱之為交換群

環 (Ring)
Def: 一集合 R，二元運算 + (加法)，與二元運算 * (乘法)，滿足：
    1. (R,+) 為交換群（加法單位元素記為 0）
    2. 乘法之封閉律：對任意 a, b ∈ R，則 a*b ∈ R
    3. 乘法之結合律：對任意 a, b, c ∈ R，則 (a*b)*c = a*(b*c)
    4. 存在乘法之單位元素：存在 1 ∈ R，使得對任意 a ∈ R，a*1 = 1*a = a
    5. 分配律：對任意 a, b, c ∈ R，則 (a+b)*c = a*c+b*c，c*(a+b) = c*a+c*b
    符合以上五點之 (R,+,*)，稱之為 "環"

交換環 (commutative ring)
Def: 一環 (R,+,*)，滿足：
    對任意 a, b ∈ R，a*b = b*a 時，
    (R,+,*) 稱之為交換環
    
體 (Field)
Def: 一集合 F，二元運算 + (加法)，與二元運算 * (乘法)，滿足：
    1. (F,+,*) 為交換環
    2. 除了 0 之外的元素均存在乘法反元素：
       對任意 a ∈ G, a != 0，存在 b ∈ G，使得 a*b = b*a = 1
    符合以上二點之 (F,+,*)，稱之為 "體"

Vector space (or linear space) 有2個operator, said + and *, ∀ x and y in V, ∃! x+y in V
for element a in F and x in V, ∃! ax in V
1. ∀ x and y in V, x+y = y+x
2. ∀ x, y and z in V, (x+y)+z = x+(y+z)
3. There exists a vector denoted 0 such that x+0 = x for each vector x in V
4. For each vector x in V, there is a vector y in V such that x+y=0
5. For each vector x in V, 1x=x
6. For each pair of real numbers a and b and each vector x in V, (ab)x=a(bx)
7. For each real number a and each pair of vectors x, y in V, a(x+y)=ax+ay
8. For each pair of real numbers a and b and each vector x in V, (a+b)x=ax+bx

   Define (a1, a2)+(b1, b2) = (a1+b1, a2-b2) and c(a1, a2)=(ca1, 0), 請問是 Vector space嗎?

   Definition: A subset W of a vector space V over a field F is called a subspace of V
   if W is a vector space over F with the operations of addition and scalar multiplication defined on V.

   Subspace
   1. X+Y ∈ W whenever x∈W and y∈W
   2. ax∈W whenever a∈F and x∈W
   3. W has a zero vector
   4. Each vector in W has an additive inverse in W.