If
you have any doubt, please comment and I will try my best to
explore this to you. Thank you.
5. Show that I = {x € R: x is irrational} does not have measure zero...
Question 5. Let x,y E R. Prove that if x and y are irrational, then at least one of 2 + y and c-y is irrational
2. Įpp. 492, Marsden & Hoffman Let y : [a,b] → R and ψ : R → R be continuous. Show that A = {(x,o(x)) : x [a,아 C R2 has volume zero in R2 and the set B-{(x, ψ (x)) : x E R} C R2 has measure zero in IK.
2. Įpp. 492, Marsden & Hoffman Let y : [a,b] → R and ψ : R → R be continuous. Show that A = {(x,o(x)) : x [a,아...
5. Define f:R + R by f(x) = x2 if x is rational, and f(x) = 0 if x is irrational. Show that f'(0) exists and is equal to zero.
4. [5 Pts] Prove that the product of a non-zero rational number and an irrational number is irrational. Can you use a direct proof? Why or why not?
(i) Show that a non-zero polynomial in ??[?]Zp[x] has exactly
?−1p−1 associates.
(ii) Let ?R be a field, 0≠?(?),?(?)∈?[?]0≠a(x),b(x)∈R[x]. Prove
that ?(?)a(x) ?(?)b(x) are associates of each other if and only if
?(?)∣?(?)a(x)∣b(x)and ?(?)∣?(?)b(x)∣a(x).
Q5 (4 points) (i) Show that a non-zero polynomial in Zp[x] has exactly p - 1 associates. a(x), b(x) E R[x]. Prove that a(x) b(x) are associates of each other if and only if a(x) | b(x) (ii) Let R be a field, 0 and b(x)...
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
5. If f :Rd + [0,0] is Lebesgue measurable, show that the Lebesgue measure of {(x, y) e Rd > R: 0 < y = f(x)} exists and equals Sed f.
Problem B. (3 pts) Show that V-5 is not a rational mumber (i.e, irrational number).
Problem B. (3 pts) Show that V-5 is not a rational mumber (i.e, irrational number).
5. Assume that we have proved that V2 and 13 are irrational numbers. Prove that 12+13 is an irrational number.
2. [pp. 492, Marsden & Hoffman] Let p [a, bR and :R R be continuous. Show that R} C R2 /-{(z, p (x)) : x e [a, b]} C R2 has volume zero in R2 and the set B-{(x, ψ(x)) : x has measure zero in R ie set F
2. [pp. 492, Marsden & Hoffman] Let p [a, bR and :R R be continuous. Show that R} C R2 /-{(z, p (x)) : x e [a, b]} C R2...