lecture 5

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