math problem!help

mitmydreamUID: 2113494

1) if A and B are events in a probability space such that 0<P(A)=P(B)=P(A交B)<1,which can not be true?
ten answer is A and B are independent. 我觉得另一个答案A is a proper subset of B也该选呀？
2）let A and B be the subsets of a set S0={A,B}.For i>=0,define Si+1inductively to be the collection of subsets X of M that are of the form C并D，C交D，M-C,where
C,D属于Si。let S=Si的并。which is the largest possible unmber of elments of S?
the answer is 16
3)let f be a real-valued function with domain [0,1].if there is some K>0such that
f(x)-f(y)<=k|x-y|, which of the following must be true?
the answer : f is continuous on (0,1),but may not be differentiable on (0,1)
4)Suppose that f is a continuous real-valued function defined on the closed interval
[0,1].
有个选项：There is a constant E>0 such that |f(x)-f(y)|<=E|x-y| for all x
and y in [0,1].为什么这个是错的？
5) what is wrong with the following argument?
let R be the real numbers, x,y 为实数
1) For all x,y, f(x)+f(y)=f(xy) is equivalent to
2) For all x,y,f(-x)+f(y)=f((-x)y) is equivalent to
3)For all x,y,f(-x)+f(y)=f((-x)y)=f(x(-y))=f(x)+f(-y)
   from this for y=0, we make the conclusion
4)For all x,f(-x)=f(x) since the steps are reversible, any fucntion with property(4)
   has property (1).therefore,for all x,y, cosx+cosy=con(xy)
a)2 does not imply 1  b)3 does not imply 2    c) 3does not imply 4
d)4does not imply 3   e) 4 is not true for f=cos
the answer is d,why?

fygoodUID: 175642

3)LIP条件不能保证可导
4)如Y=X^(1/(1-x))这里只有导数有界才行
5)就是说f(-x)+f(y)=f((-x)y)是不对的!