9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
0 and 0, and let a E Z. Prove that [a],m C [a]n if and only if n | Let m,EN with m TT 0 and 0, and let a E Z. Prove that [a],m C [a]n if and only if n | Let m,EN with m TT
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...
Let 1 ≤ m ∈ Z and let a,b ∈ Z be such that gcd(a,b) = m. Prove that gcd(a2,b2) = m2.
2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n 2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n
2. Fix m, n E N. Define a mapping f:Z/nZ+Z/mZ by f([a]n) = [a]m. a. Prove that if m | n then f is a well-defined function. That is, prove that if (a)n = [b]n then f([a]n) = f([b]n). b. Let n = 12 and m = 3. Write PreImp({[1]3, [2]3}) in roster notation. c. Suppose mfn. Show that f is ill-defined. That is, show there exist a, b E Z such that (a)n = [b]n but f([a]n) + f([b]n).
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers. Let s,t E Z be such that sm+tn The Chinese Remaindering Theorem states that for every a, b E Z there exists c E Z such that r a mod m (Va E Z) b mod nmod mn (3) where a convenient c is given by 1. Prove that the above c satisfies both ca mod m and cb mod n 2. LetxEZ. Prove...
fekri/n k0,1,...,n-1}, called the nth roots of unity. A primitive root of unity is = eri/n for which 2. The roots off(x) = x"-1 are the n complex numbers Cn and are ged(n, k) 1. It is easy to see that Q(C) is the splitting field of zn - 1. (a) For each n 3,... ,8, sketch the nth roots of unity in the complex plane. Use a different set of axes for each n. Next to each root, write...
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a) If b | c, then gcd(a, b) gcd(a, c). (b) If b c, then ged(a., b) < gcd(a, c)
Advanced Calculus (3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion. (3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....