8. (10 points) If A is a real symmetric matrix, find a matrix V such that VAVT = I. You need to prove your answer is correct.

(b) (4 points) For each pair of sets below, determine if |A| ＜ |B|, |A| = |B|, or |A| ＞ |B| (N.B. N is the set of natural numbers.) 1) A = The set of all computable functions from N to N. B = The set of all uncomputable functions from N to N. 2) A = The set of all C++ programs that terminate. B = The set of all C++ programs that don't terminate