Ans>option 1.

QUESTION 14 Define g:Z → Z by the rule g(n)= n2 - 2, for all integers n. Is gone-to-one? Yes, because Vm, nez, if g(m) = g(n) then m=n. Yes, because 1, -1€ Z and g(1)-(-1). No. A counterexample would be g(1)-(-1) but 1-1. No. A counterexample would be g(1) g(0) and 1=0.
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.
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)
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...
(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...
8. Define (n) to be the number of positive integers less than n and n. That is, (n) = {x e Z; 1 < x< n and gcd(x, n) = 1}|. Notice that U (n) |= ¢(n). For example U( 10) = {1, 3,7, 9} and therefore (10)= 4. It is well known that (n) is multiplicative. That is, if m, n are (mn) (m)¢(n). In general, (p") p" -p Also it's well known that there are relatively prime, then...