1. Recall we define multiplication in Zn by [a] :n [b] = [ab]. Prove that this...
4. (a) For n eZ, define multiplication mod n by ao b-a b (where indicates regular real number multiplication), prove that On is a binary operation on Zn. That is, (Hint your proof will be very similar to the proof for homework 4 problem 7ab) (b) Let n E Z. Is the binary algebraic structure 〈L,On) always a group? Explain. (c) Prove There exists be Zn such that a n I if and only if (a, n)1. (d) It is...
10.) Consider (Zn; n), the set Zn with mod n multiplication. i. Argue that if neither a nor b has any common divisors greater than 1 with n then neither does ab. [Equivalently gcd(a; n) = 1, etc.] ii. Argue that if a does not have any common divisors greater than 1 with n, then [a]n has a multiplicative inverse in Zn. iii. Argue that (i) and (ii) imply that the set of elements f[a]n 2 Znjgcd(a; n) = 1g...
Im wondering how to do b). (6) We define the set of compactly supported sequences by qo = {(zn} : there exists some N > 0 so that Zn = 0 for all n >N). We define the set of compactly supported rational sequences by A={(za) E ao : zn E Q for all n E N). (a) Prove that A is countable (b) Prove that for 1 S p<oo the set A is dense in P. You may use...
Recall that is the group of units in , with operation given by multiplication. Consider the function defined by . Show that this function is well-defined. Show that is an isomorphism. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Please answer all parts. Thank you! 20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...
Let ne N. Show that Zn forn > 2 is not a group under multiplication as defined above. What happens for n = 1?
9 is a ka. Prove that For ke Z,, define a map k : Zn homomorphism. Zn by a 10 Prove that kis an isomorphism if and only if k is a generator of Zn. Show that every automorphism of Z, is of the form k, wherek is a generator of Zn. Aut(Zn) is an isomorphism, where y : k pk 1 Prove that y: U(n)
2) Let X,..X, be ii.d. N(O, 1) random variables. Define U- Find the limiting distribution of Zn (Hint: Recall that if X and Y are independent N(0, 1) random variables, then has a Cauchy distribution 2) Let X,..X, be ii.d. N(O, 1) random variables. Define U- Find the limiting distribution of Zn (Hint: Recall that if X and Y are independent N(0, 1) random variables, then has a Cauchy distribution
QUESTION 8 Let G is an abelian group with the additive operation. Define the operation of multiplication by the rule ab- a- b for all a,b of G. Is G a ring? QUESTION 8 Let G is an abelian group with the additive operation. Define the operation of multiplication by the rule ab- a- b for all a,b of G. Is G a ring?