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
Homework Answers
Add Answer to:
a) Prove that Axiom (D1) holds for Z/nZ.
Here is D1 that is needed:
D1) For...
(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...
Exercise 4.1: Explain/prove why the following sets and binary operations do not define groups (so just...
Exercise 4.1: Explain/prove why the following sets and binary operations do not define groups (so just try to determine one group axiom that fails to hold): 1) The set of polynomials of odd degree under addition. 2) The set of polynomials of odd degree under multiplication 3) The set of integers congruent to 1 modulo 11, under addition modulo 11. 4) The set of integers modulo 11, under multiplication modulo 11. 5) The set of nonzero integers modulo 4, under...
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...
(6 pts) Alternate construction of the integers from the natural numbers. Suppose that the natural numbers...
(6 pts) Alternate construction of the integers from the natural numbers. Suppose that the natural numbers N = {0,1,2, ...} ations. We do not yet have a notion of subtraction or the cancellation law for addition (if x+y = x+ z, then y = 2) and for multiplication given with the usual addition and multiplication oper negative numbers, though we do have are Define a relation R on N2 as follows (a, b) R (c, d) if and only if...
8.) Consider the integers Z. Dene the relation on Z by x y if and only...
8.) Consider the integers Z. Dene the relation on Z by x y if
and only
if 7j(y + 6x). Prove:
a.) The relation is an equivalence relation.
b.) Find the equivalence class of 0 and prove that it is a subgroup
of Z
with the usual addition operator on the integers.
8.) Consider the integers Z. Define the relation ~ on Z by x ~ y if and only if 7)(y + 6x). Prove: a.) The relation is an...
In the video I showed that z/nZ satisfies the left distributive law. In this Qwickly Jot...
In the video I showed that z/nZ satisfies the left distributive law. In this Qwickly Jot assignment I want you to verify that ZinZ satifies the right distributive law. In other words, verify that (a + n 2) = (b + n 2)) (c + n Z) = (a + n 2) (c + n 2)) e (b+ n 2) ® (a + n 2)) sor as Part 1: Write out the addition and multiplication tables for Z/6Z Part 2:...
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...
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).
(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 :=...
(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...
Exercise 4.1: Explain/prove why the following sets and binary operations do not define groups (so just try to determine one group axiom that fails to hold): 1) The set of polynomials of odd degree under addition. 2) The set of polynomials of odd degree under multiplication 3) The set of integers congruent to 1 modulo 11, under addition modulo 11. 4) The set of integers modulo 11, under multiplication modulo 11. 5) The set of nonzero integers modulo 4, under...
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...
(6 pts) Alternate construction of the integers from the natural numbers. Suppose that the natural numbers N = {0,1,2, ...} ations. We do not yet have a notion of subtraction or the cancellation law for addition (if x+y = x+ z, then y = 2) and for multiplication given with the usual addition and multiplication oper negative numbers, though we do have are Define a relation R on N2 as follows (a, b) R (c, d) if and only if...
8.) Consider the integers Z. Dene the relation on Z by x y if
and only
if 7j(y + 6x). Prove:
a.) The relation is an equivalence relation.
b.) Find the equivalence class of 0 and prove that it is a subgroup
of Z
with the usual addition operator on the integers.
8.) Consider the integers Z. Define the relation ~ on Z by x ~ y if and only if 7)(y + 6x). Prove: a.) The relation is an...
In the video I showed that z/nZ satisfies the left distributive law. In this Qwickly Jot assignment I want you to verify that ZinZ satifies the right distributive law. In other words, verify that (a + n 2) = (b + n 2)) (c + n Z) = (a + n 2) (c + n 2)) e (b+ n 2) ® (a + n 2)) sor as Part 1: Write out the addition and multiplication tables for Z/6Z Part 2:...
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...
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).
(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 :=...
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