Math 74 Final ExamDecember 16th, 2008Name SIDQuestion Score Possible1 102 113 64 65 106 67 58 7P6111. (a) Let (X, d) be a metric space, and let (xn) be a sequence in (X, d).Define what it means for (xn) to be convergent.(b) Use quantifier negation to give a definition of “(xn) is not con-vergent.”(c) Pick your favorite metric space (X, d) and your favorite non-convergent sequence (xn) in (X, d). Use your definition from part(b) to prove that your sequence (xn) is not convergent.22. Let X be a set. We call a function d : X × X → R a pseudometric ifthe following hold:(a) For all x ∈ X, we have d(x, x) = 0.(b) For all x, y, ∈ X, we have d(x, y) = d(y, x).(c) For all x, y, z ∈ X we have d(x, y) ≤ d(x, z) + d(z, y).Let d be a pseudometric on X. Do the following:(a) Explain how this definition differs from the definition of a metric.(b) Show that d(x, y) ≥ 0 for all x, y ∈ X.(c) Let x, y ∈ X. Show that if d(x, y) = 0 then d(x, z) = d(y, z) forall z ∈ X.(d) Show that the relation ∼ on X given by x ∼ y iff d(x, y) = 0 isan equivalence relation.3(Additional Space for Work on Problem 2)43. Show that there does not exist a rational number q such that q3= 2.54. Show thatnXi=11i(i + 1)=nn + 1for all n ∈ N.65. Let ρ be the discrete metric on R, let d be the Euclidean metric on R,and letf : (R, ρ) → (R, d)be the function defined by f(r) = r for all r ∈ R.(a) Is f continuous? Prove your answer.(b) Is f−1continuous? Prove your answer.76. Let A ⊆ R be a closed subset of R in the Euclidean metric. Show thatA × {0} ⊆ R2is a closed subset of R2in the Euclidean metric.87. Do ONE of the following:(a) Show that there are infinitely many primes p ∈ N such that p + 2is also a prime.(b) Show that every even n ∈ N with n > 2 can be written as n = p+qfor some primes p and q.(c) Let p ∈ N be a prime and let a ∈ Z be arbitrary. Show that ifa2≡ 1(mod p) then either a ≡ 1(mod p) or a ≡ −1(mod p).98. Write a short (≤ 2 page) essay entitled “Using equivalence relationsto construct new mathematical objects.” You do not need to includeproofs, and your essay should look like an “English class essay,” i.e.should be in usual paragraph format. Include examples from class toillustrate your points.10(Additional Room For Solutions)11(Additional Room For