Tutorial ex. 3: Prove that the subgroups of Z/NZ are exactly given by <a mod N>,...
Tutorial ex. 3: Prove that the subgroups of Z/NZ are exactly given by <a mod N>, for a | N.
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Tutorial ex. 3: Prove that the subgroups of Z/NZ are exactly
given by <a mod N>,...
(2) For an integer n, let Z/nZ denote the set of equivalence classes [k) tez: k -é is divisible by n (a) Prove that the set Z/nZ has n elements. (b) Find a minimal set of representatives for these...
(2) For an integer n, let Z/nZ denote the set of equivalence classes [k) tez: k -é is divisible by n (a) Prove that the set Z/nZ has n elements. (b) Find a minimal set of representatives for these n elements. (c) Prove that the operation gives a well-defined addition on Z/nZ Hint: The operution should not depend on the choice of coset representatives Verify that this gives Z/n2 the structure of an ahelian group. Be sure to verify all...
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × ...
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × Z/nZ whith f(x) = (x + mZ, x + nZ) is surjective (b) Prove the Chinese Remainder Theorem: If m and n are relatively prime integers > 1 and if a and b are any integers, then there exists a E Z such that b(mod n). a(mod m) and a a Hint: (a)]
27. (a) Let...
2. Fix m, n E N. Define a mapping f:Z/nZ+Z/mZ by f([a]n) = [a]m. a. Prove...
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).
Prove that for all integers n, (-n) mod 2 = n mod 2. Give an example...
Prove that for all integers n, (-n) mod 2 = n mod 2. Give an example to show that it is not always true that (-n) mod 3 = n mod 3. Professor mentioned to prove for odd and even integers, however, I don't know how to start the proof.
a) Prove that Axiom (D1) holds for Z/nZ. Here is D1 that is needed: D1) For...
a) Prove that Axiom (D1) holds for Z/nZ.
Here is D1 that is needed:
D1) For all a, b, c ∈ R, a · (b + c) = a · b + a · c
(10 pts) Let n be a natural number and consider the set Z/nZ of equiva- lence classes of integers modulo n. Define addition and multiplication as on the equivalence class worksheet.
= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by...
= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer (b) (5 Pts) Prove that o is surjective onto its image. Answer
= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer
(b) (5 Pts) Prove that o is surjective onto its image. Answer
Prove that Z/ ≡3 has exactly three elements using the given hint! Definition: Let R be...
Prove that Z/ ≡3 has exactly three elements using the
given hint!
Definition: Let R be an equivalence relation on the set A. The set of all equivalence classes is denoted by A/R (g) Prove that Z/ has exactly three elements. Hint: First, verify that [5]3, [7]3, and [013 are three different elements of Z/-3-Then, verify that every m E Z is in one of these sets. Then explain why those two facts imply that [5]3, [7 3, and [013...
Elucidean Algorithm. Q 4. For each of the following equations, find a solution z Z or prove that no solution z E Z exists (a) 7x 13 mod 83 mod 624; 11 (b) 25x + 3 (c) 36r 1 mod 87. 12 marks In all...
Elucidean Algorithm.
Q 4. For each of the following equations, find a solution z Z or prove that no solution z E Z exists (a) 7x 13 mod 83 mod 624; 11 (b) 25x + 3 (c) 36r 1 mod 87. 12 marks In all cases, explain your reasoning.
Q 4. For each of the following equations, find a solution z Z or prove that no solution z E Z exists (a) 7x 13 mod 83 mod 624; 11 (b)...
Prove the following: a. a = b (mod b) implies b =a (mod n) a =...
Prove the following: a. a = b (mod b) implies b =a (mod n) a = b (mod n) and b = c (mod n) imply a = c(mod n)
Prove (X - Y) nZ = (X n 2) - Y
Prove (X - Y) nZ = (X n 2) - Y
(2) For an integer n, let Z/nZ denote the set of equivalence classes [k) tez: k -é is divisible by n (a) Prove that the set Z/nZ has n elements. (b) Find a minimal set of representatives for these n elements. (c) Prove that the operation gives a well-defined addition on Z/nZ Hint: The operution should not depend on the choice of coset representatives Verify that this gives Z/n2 the structure of an ahelian group. Be sure to verify all...
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × Z/nZ whith f(x) = (x + mZ, x + nZ) is surjective (b) Prove the Chinese Remainder Theorem: If m and n are relatively prime integers > 1 and if a and b are any integers, then there exists a E Z such that b(mod n). a(mod m) and a a Hint: (a)]
27. (a) Let...
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).
a) Prove that Axiom (D1) holds for Z/nZ.
Here is D1 that is needed:
D1) For all a, b, c ∈ R, a · (b + c) = a · b + a · c
(10 pts) Let n be a natural number and consider the set Z/nZ of equiva- lence classes of integers modulo n. Define addition and multiplication as on the equivalence class worksheet.
= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer (b) (5 Pts) Prove that o is surjective onto its image. Answer
= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer
(b) (5 Pts) Prove that o is surjective onto its image. Answer
Prove that Z/ ≡3 has exactly three elements using the
given hint!
Definition: Let R be an equivalence relation on the set A. The set of all equivalence classes is denoted by A/R (g) Prove that Z/ has exactly three elements. Hint: First, verify that [5]3, [7]3, and [013 are three different elements of Z/-3-Then, verify that every m E Z is in one of these sets. Then explain why those two facts imply that [5]3, [7 3, and [013...
Elucidean Algorithm.
Q 4. For each of the following equations, find a solution z Z or prove that no solution z E Z exists (a) 7x 13 mod 83 mod 624; 11 (b) 25x + 3 (c) 36r 1 mod 87. 12 marks In all cases, explain your reasoning.
Q 4. For each of the following equations, find a solution z Z or prove that no solution z E Z exists (a) 7x 13 mod 83 mod 624; 11 (b)...
Prove the following: a. a = b (mod b) implies b =a (mod n) a = b (mod n) and b = c (mod n) imply a = c(mod n)
Prove (X - Y) nZ = (X n 2) - Y
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