Suppose that 2- {1,2, with P()0 and P1,2)-1 Suppose P{1) = 1. Prove that P is...
p-1 mod 4, prove that Σ k ( )-0. Let p be an odd prıme. Suppose that p k=1
p-1 mod 4, prove that Σ k ( )-0. Let p be an odd prıme. Suppose that p k=1
2 x Problem 2. Prove that f(x) = is a bijection frorn [1,2] to [0, 1].
Let 2 [0, 1], and let F be the collection of every subset of such that the subset or its complement is countable. Let P(.) be a measure on F such that for A E F, P(A) if A is countable and P(A)1 if Ac is countable. (a) Is F a field? Also, is F a σ-field? (Note that afield is closed under finite union while a σ-field is closed under countable union. (b) Is P finitely additive? Also, is...
A . Prove that Problem 4. (2 points) Let A and B be two sets. Suppose that A B = B A = B. Problem 5. (optional but recommended). Show that the set X = {(...) 21: sequences of O's and I's is not countably infinite. Hint: think of a natural function between X and P(N). € {0,1}} of infinite
(3) Suppose that f E C'((0, 1]). Given e > 0, prove that there exists a polynomial p such that lf-plloo -p'| <E
(3) Suppose that f E C'((0, 1]). Given e > 0, prove that there exists a polynomial p such that lf-plloo -p'|
2. Suppose P and Q are positive odd integers such that (PQ)-1. Prove that Qm] Pn] P-1 0-1 0<m<P/2 0<n
g(p+1)/2 (a) Suppose 9 is a p rimitive root of an odd prime p. Prove that- (mod p)
g(p+1)/2 (a) Suppose 9 is a p rimitive root of an odd prime p. Prove that- (mod p)
Suppose that pı, P2, ..., P, are the only primes congruent to 1 (mod 4). Prove that 4p?p, ... p, + 1 is divisible only by primes congruent to 3 (mod 4). Assuming that all odd prime factors of integers of the form x2 +1 are congruent to 1 (mod 4), use Exercise 6 to prove that there exist infinitely many primes congruent to 1 (mod 4).
8. Suppose V is an n-dimensional complex vector space. Suppose T E C(V) is such that 1,2, and 3 are the only distinct eigenvalues of T (a) Prove that the dimension of each generalized eigenspace of T is at most (n - 2). (b) Show that (T-1)"-2(T-21)"-"(7-31)"-"(a) = 0V, for all α є V.
8. Suppose V is an n-dimensional complex vector space. Suppose T E C(V) is such that 1,2, and 3 are the only distinct eigenvalues of T...
(2) Prove that if j-0 i-0 with k, 1 e N u {0), and bo, . . . , be , do, . . . , dl e { 0, . . . , 9), such that be, de # 0, then k = 1 and bi- di fori 0,.. , k. (I recommend using strong induction and uniqueness of the expression n=10 . a + r with a e Z and re(0, 1, ,9).) (3) Prove that for all...