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Problem 7. Fix a natural number n € N, and let en denote the equivalence relation...
6. Fix n E N and recall the definition of the equivalence relation on Z given...
6. Fix n E N and recall the definition of the equivalence relation on Z given by a = b mod n. (This means that a – b = kn, for some k € Z.) Let [a] denote the equivalence class containing a. (a) Show that defining [a] + [b] := (a + b] makes sense, i.e. does not depend on the choice of representatives for the classes. (b) Show that defining [a] × [b] := [a x b] makes...
probelms 9.1 9 Modular arithmetic Definition 9.1 Let S be a set. A relation R =...
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
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...
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...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k,...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
(i) Prove that the realtion in Z of congruence modulo p is an equivalence relation. Namesly,...
(i) Prove that the realtion in Z of congruence modulo p is an equivalence relation. Namesly, show that Rp := {(a,b) € ZxZ:a = 5(p)} is reflexive, symmetric and transitive. (ii) Let pe N be fixed. Show that there are exactly p equivalence classes induced by Rp. (iii) Consider the relation S E N defined as: a Sb if and only if a b( i.e., a divides b). Prove that S is an order relation. In other words, S :=...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92)gig2. Let X be a set. An action ofG on X is a a map that satisfies the following tuo conditions: c. Let G be a finite group. For each E X, consider the map (aje- fer all elements r X (b) 9-(92-2) for all 91,92 G and all r E X Show that is surjective and that, for all y...
2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tive...
number thoery
just need 2 answered
2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tively prime to n by p(n). Let a, b be positive integers such that ged(a,n) god(b,n)-1 Consider the set s, = {(a), (ba), (ba), ) (see Prollern 1). Let s-A]. Show that slp(n). 1. Let a, b, c, and n be positive integers such that gcd(a, n) = gcd(b, n) = gcd(c, n) = 1 If...
Problem 7. (20 pts) Let n EN be a natural nmber and X a finite set...
Problem 7. (20 pts) Let n EN be a natural nmber and X a finite set with n elements. Show that the number of permutations of X such that no element stays in the same position is n! k! k o For instance, there are 6 = 3! permutations of 3 elements, but only 2 of them are permutations which fix no element. Similarly, there are 24-4! permutations of 4 elements, but only 9 which fix no element. Hint: Use...
Just number 5 please 4. Show that congruence modulo n is an equivale 5. Consider the...
Just number 5 please
4. Show that congruence modulo n is an equivale 5. Consider the unit circle C with equation modulo n is an equivalence relation UI arcle C with equation x2 + y2 = 1 in the plane. Thus C = {(x, y) ERX R: x2 + y2 = 1}. Define a relation on C as follows: For (x, y) EC, antipodal point (-x, -y). In symbols, llows: For (x, y) E C (x, y) is related to...
I. Let each of R, S, and T be binary relations on N2 as defined here: R-[<m, n EN nis the smallest prime number greater than or equal to m] S -[< m, n> EN* nis the greatest prime number less...
I. Let each of R, S, and T be binary relations on N2 as defined here: R-[<m, n EN nis the smallest prime number greater than or equal to m] S -[< m, n> EN* nis the greatest prime number less than or equal to m] (a) Which (if any) of these binary relations is a (unary) function? (b) Which (if any) of these binary relations is an injection? (c) Which (if any) of these binary relations is a surjection?...
6. Fix n E N and recall the definition of the equivalence relation on Z given by a = b mod n. (This means that a – b = kn, for some k € Z.) Let [a] denote the equivalence class containing a. (a) Show that defining [a] + [b] := (a + b] makes sense, i.e. does not depend on the choice of representatives for the classes. (b) Show that defining [a] × [b] := [a x b] makes...
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
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...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
(i) Prove that the realtion in Z of congruence modulo p is an equivalence relation. Namesly, show that Rp := {(a,b) € ZxZ:a = 5(p)} is reflexive, symmetric and transitive. (ii) Let pe N be fixed. Show that there are exactly p equivalence classes induced by Rp. (iii) Consider the relation S E N defined as: a Sb if and only if a b( i.e., a divides b). Prove that S is an order relation. In other words, S :=...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92)gig2. Let X be a set. An action ofG on X is a a map that satisfies the following tuo conditions: c. Let G be a finite group. For each E X, consider the map (aje- fer all elements r X (b) 9-(92-2) for all 91,92 G and all r E X Show that is surjective and that, for all y...
number thoery
just need 2 answered
2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tively prime to n by p(n). Let a, b be positive integers such that ged(a,n) god(b,n)-1 Consider the set s, = {(a), (ba), (ba), ) (see Prollern 1). Let s-A]. Show that slp(n). 1. Let a, b, c, and n be positive integers such that gcd(a, n) = gcd(b, n) = gcd(c, n) = 1 If...
Problem 7. (20 pts) Let n EN be a natural nmber and X a finite set with n elements. Show that the number of permutations of X such that no element stays in the same position is n! k! k o For instance, there are 6 = 3! permutations of 3 elements, but only 2 of them are permutations which fix no element. Similarly, there are 24-4! permutations of 4 elements, but only 9 which fix no element. Hint: Use...
Just number 5 please
4. Show that congruence modulo n is an equivale 5. Consider the unit circle C with equation modulo n is an equivalence relation UI arcle C with equation x2 + y2 = 1 in the plane. Thus C = {(x, y) ERX R: x2 + y2 = 1}. Define a relation on C as follows: For (x, y) EC, antipodal point (-x, -y). In symbols, llows: For (x, y) E C (x, y) is related to...
I. Let each of R, S, and T be binary relations on N2 as defined here: R-[<m, n EN nis the smallest prime number greater than or equal to m] S -[< m, n> EN* nis the greatest prime number less than or equal to m] (a) Which (if any) of these binary relations is a (unary) function? (b) Which (if any) of these binary relations is an injection? (c) Which (if any) of these binary relations is a surjection?...
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