1.
When 2. m * n = n * m is not always true, such as m, n for the array
3. u is the universal acronym, that Complete Works
4. proven formula in strict accordance with the order of writing, often need to use commutativity reverse the order, each step can only use a formula
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
A proof to prove another
7. binary relations
a binary relation between S and T is a subset of S*T
8. binary relation defined
a. listed directly {(1,1), (2,3), (3,2)}
. B lists the ranges {(x, y) ∈ [1,3] × [1,3]: 5 | xy -1}
c. Release still other relation {(1,1)} ∪ {(2,3)} ∪ {(2,3)} ←
d.
e.
f.
9. binary relation properties
a. (R) re fl exive associated with itself, such as equal to
For all x ∈ S: (x,x) ∈ R
b. (AR) antire fl exive not related to itself, such as less than
For all x ∈ S: (x,x) / ∈ R
c. (S) symmetric if not equal
For all x,y ∈ S: If (x,y) ∈ R then (y,x) ∈ R
d. (AS) antisymmetric such as less than
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