1. S is a compact connected set in R^n, and how many connected components can the complement of S have?

I totally don't know how to deal with this.....

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2. X is a set, f: X->P(x) [power set of X] is a function. Z={s: s belongs to X, and s doesn't belong to f(X)}

A. Z is empty

B. Z is not empty

C. Z=X

D. Z doesn't belong to P(X)

E. the complement of Z belongs to P(X)

I think the answer might be B, coz f(X) is a set of set..

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3. Which are equal to "map f is continuous"?

I. For every set A, f^(-1)(in(A))=in(f^(-1)(A))

II. For every set B, cl(f^(-1)(B)) contains f^(-1)(cl(B))

III. f is a open map

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4. a,b belong to group G, both have finite orders

I. If ab=ba, then ab has finite order

II. If ab has finite order,then ba has finite order

III. If ab has finite order, then a^(-1)b^(-1) has finite order

still Which are correct?....

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5.it is a linear algebra question.

Tu=u,Tv=2v,Tw=3w. T is the linear transformer on R^3. which are right?

I. detT=6;

II. u,v,w is a basis of R^3

III. character poly |xI-T|=(x-1)(x-2)(x-3)

I think all of them are correct, how about you?....

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6. T is an orthogonal matrix, which of the following is right?[. means dot product]

I. Tu . Tv = u . v

II. If v . (Sqrt[2]/2,Sqrt[2]/2,0)=0, Tv . v=0

III. T^2=1

A friend think they are all right.

But i think T is not a normed orthogonal matrix, coz i remember only orthogonal property cannot ensure its determinant is 1.

Ironically, http://en.wikipedia.org/wiki/Orthogonal_matrix

orthogonal matrix has been normed from wiki. Am I wrong?...

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thanks, and please enlighten me before my test.....=_____=