Fourth Homework Set Answers

5. (a) dim(A ∪ B) = max{dim(A), dim(B)} = max{1.585, 0.631} = 1.585

(b) dim(B ∪ C) = max{dim(B), dim(C)} = max{0.631, 2} = 2

(c) dim(A × B) = dim(A) + dim(B) = 1.585 + 0.631 = 2.216

(d) dim(B × C) = dim(B) + dim(C) = 0.631 + 2 = 2.631

(e) codim(A ∩ C) = codim(A) + codim(C). That is, 3 - dim(A ∩ C) = 3 - dim(A) + 3 - dim(C), or

dim(A ∩ C) = dim(A) + dim(C) - 3

So dim(A ∩ C) = 1.585 + 2 - 3 = 0.585

(f) codim(B ∩ C) = codim(B) + codim(C), so

dim(B ∩ C) = dim(B) + dim(C) - 3

This gives dim(B ∩ C) = 0.631 + 2 - 3 = -0.369

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