

QUESTION 14 Define g:Z → Z by the rule g(n)= n2 - 2, for all integers...
Define g:Z → Z by the rule g(n)= n2 – 2, for all integers n. is gone-to-one? No. A counterexample would be g(1)= g(-1) but 17 -1. Yes, because 1, -1€ Z and g(1)= g( - 1). No. A counterexample would be g(1)+g(0) and 1 +0. Yes, because Vm, ne Z, if g(m)= g(n) then m=n.
descrite math
Define q, r, and s, all functions on the integers, by q(n) = n2 + 3", r(n) = n +3, and s(n) = 7n -5. Determine: a. qºros b. rosoa c. (qºsor (2) d. (soroq)(3)
4) Let D be the set of all finite subsets of positive integers. Define a function (:2 - D as follows: For each positive integer n, f(n) =the set of positive divisors of n. Find the following f (1), f(17) and f(18). Is f one-to-one? Prove or give a counterexample.
Problem 4. Show that for all integers n, n2 mod 3- 1 n2 mod 3- 0 or (i.e. there exists an integer k such that n2 3k or n2 3k +1). mp
Let X0,X1,... be a Markov chain whose state space is Z (the
integers).
Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1
= in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following
always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0)
?
(Prove if “yes”, provide a counterexample if “no”)
Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli
5. Let Zli_ {a + bi l a,b...
(14) Let R be a relation on the integers defined by m R n if and only if m+m2 n+ n2(mod 5). Show that R is an equivalence relation and determine all the equivalence classes.
(A and C)
Exercise 1.14. If n and k are integers, define the binomial coeffi- cient (m), read n choose k, by n! if 0 <k <n, = 0 otherwise. k!(n - k)! (a) Prove that ("#") = (m) + (-2) for all integers n and k. (b) By definition, () = 1 if k = 0 and 0 otherwise. The recursion relation in (a) gives a computational procedure, Pascal's triangle, for calculating binomial coefficients for small n. Start with...
2. (a) Prove the product rule for complex functions. More specifically, if f(z) and g(z) f(z)g(z) is also analytic, and that analytic prove are that h(z) h'(z)f(z)9() f(z)g'(z) (You may use results from the multivariable part of the course without proof.) = nz"- for n e N = {1,2,3,...}. Your textbook establishes that S z"= dz (b) Let Sn be the statement is true. With the help of (a), show that if Sn is true, then Sn+1 is true. Why...
Please answer all!!
3. Show that if n e Z so that n is odd then 8|n2 + (n + 6)2 +6. 4. (a) Let a, b, and n be integers so that n > 2. Define: а is congruent to b mod n. The notation here is a = b (modn). (b) Is 12 = 4 (mod 2)? Explain. (c) Is 25 = 3 (mod 2)? Explain. (d) Is 27 = 13 (mod8)? Explain. (e) Find 6 integers x...