1.
2. m*n=n*m并不总是成立，如m,n为array时
3. u为universal的缩写，即全集
4. 证明时严格按照公式顺序写，很多时候需要用commutativity调换顺序，每一步只能用一个公式
5.
6. If A is a set defined using ∩, ∪, ∅ and U, then dual(A) is the expression obtained by replacing ∩ with ∪ (and vice-versa) and ∅ with U (and vice-versa).
Absorption law: A∪(A∩B) = A
Dual: A∩(A∪B) = A
证明一个即可证明另一个
7. binary relations
a binary relation between S and T is a subset of S*T
8. binary relation的定义
a. 直接列出 {(1,1), (2,3), (3,2)}
b. 列出范围{(x,y)∈ [1,3]×[1,3] : 5|xy −1}
c. 又其他relation推出{(1,1)}∪{(2,3)}∪{(2,3)}←
d.
e.
f.
9. binary relation性质
a. (R) reflexive与自身相关，如等于
For all x ∈ S: (x,x) ∈ R
b. (AR) antireflexive不与自身相关，如小于
For all x ∈ S: (x,x) / ∈ R
c. (S) symmetric 如不等于
For all x,y ∈ S: If (x,y) ∈ R then (y,x) ∈ R
d. (AS) antisymmetric 如小于
For all x,y ∈ S: If (x,y) and (y,x) ∈ R then x = y
e. (T) transitive
For all x,y,z ∈ S: If (x,y) and (y,z) ∈ R then (x,z) ∈ R
对于含有if的定义，如果任何情况下if都不满足，则依旧成立