Let (d , d2,...,d,) be a non-increasing sequence of nonnegative integers. Prove that there exists a loopless graph with degree sequence (d ,d2...., dn) if and only if n n d, is even and d, Ed. i=1 i=2
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove that for any A E A f du lim fn du A 4 (You must show that the integrals exist.) (3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove...
The symbol N denotes the nonnegative integers, that is, N= {0,1,2,3,...}. The symbol R denotes the real numbers. In each of the proofs by induction in problems (2), (3), and (4), you must explicitly state and label the goal, the predicate P(n), the base case(s), the proof of the base case(s), the statement of the inductive step, and its proof. Your proofs should have English sentences connecting and justifying the formulas. As an example of the specified format, consider the...
5*. Consider all sequences (ai,. .., an) such that a, are nonnegative integers and a ai+ 2. Let P, n and Rn be the number of such sequences which start from 0, 1 and 2 respectively. (a) Compute P, Qn, Rn by writing down all such sequences for n 1,2,3. (b) Prove that P, Qn Rn satisfy the recurrence relations: (c) Translate the above equations into linear equations for the generating functions for P, Qn, Rn (d) Solve these equations...
Let n be a nonnegative integer and let F 22 + 1 be a Fermat number. Prove that if 3 od F., then F, is a prime number. (Note: This yields a primality test known as Pepin's Test.) Let n be a nonnegative integer and let F 22 + 1 be a Fermat number. Prove that if 3 od F., then F, is a prime number. (Note: This yields a primality test known as Pepin's Test.)
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm we can write m=qn+r, with 0 <r<n-1. As in class define (m,n) = {mc+ny: I,Y E Z} and S(..r) = {nu+ru: UV E Z}. Prove that (m,n) = S(n,r). (Remark: If we add to the definition of ged that gedan, 0) = god(0, n) = n, then this proves that ged(m, n) = ged(n,r). This result leads to a fast algorithm for computing ged(m,...
Let n be a nonnegative integer and let F 22 + 1 be a Fermat number. Prove that if is a prime number, then either n=0 or 3--1mod F. [Hint: If n 2 1, use the law of quadratic reciprocity to evaluate the Legendre symbol (3/F). Now use Euler's Criterion (Theorem 4.4).] Let n be a nonnegative integer and let F 22 + 1 be a Fermat number. Prove that if is a prime number, then either n=0 or 3--1mod...
Problem statement: Prove the following: Theorem: Let n, r, s be positive integers, and let v1, . . . , vr E Rn and wi, . . . , w, є Rn. If wi є span {v1, . . . , vr} for each i = 1, . . . , s, then spanfVi, . .., v-) -spanfvi, . .., Vr, W,...,w,) Suggestiorn: To see how the proof should go, first try the case s - 1, r 2..] Problem...
13.12.8 Problem. Let R be a ring and, let M be an R-module. Let m be a nonnegative integer, and suppose that M1,..., Mm are R-submodules of M, and that M is the internal direct sum of M1,..., Mm. Let n be a nonnegative integer with n < m, and for each i E {1,...,n}, let N; be an R-submodule of M. Let N = N1 ++ Nn. ... (i) Prove that N is the internal direct sum of N1,...,...
Let a and b be nonnegative integers. Write a complete proof for the fact that a + b = 0 implies a = 0.