Let S = {(a1,a2,...,an)|n ≥ 1,ai ∈ Z≥0 for i = 1,2,...,n,an ̸=
0}. So S...
Let S = {(a1,a2,...,an)|n ≥ 1,ai ∈ Z≥0 for i = 1,2,...,n,an ̸=
0}. So S is the set of all finite ordered n-tuples of nonnegative
integers where the last coordinate is not 0. Find a bijection from
S to Z+.
Prove that a countable union of countable sets countable; i.e.,
if {Ai}i∈I is a collection of...
Prove that a countable union of countable sets countable; i.e.,
if {Ai}i∈I is a collection of sets, indexed by I ⊂ N, with each Ai
countable, then union i∈I Ai is countable. Hints: (i) Show that it
suffices to prove this for the case in which I = N and, for every i
∈ N, the set Ai is nonempty. (ii) In the case above, a result
proven in class shows that for each i ∈ N there is a...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....
Consider the ring R = Z ∞ = {(a1, a2, a3, · · ·) : ai...
Consider the ring R = Z ∞ = {(a1, a2, a3, · · ·) : ai ∈ Z for
all i}. It turns out that R forms a ring under the operations (a1,
a2, a3, · · ·) + (b1, b2, b3, · · ·) = (a1 + b1, a2 + b2, a3 + b3,
· · ·), (a1, a2, a3, · · ·) · (b1, b2, b3, · · ·) = (a1 · b1, a2 ·
b2, a3 ·...
(8 marks)
Let S = {(a1, a2, . . . , an)| n ≥ 1, ai...
Let S = {(a1, a2, . . . , an)| n ≥ 1, ai ∈ Z ≥0 for i = 1, 2, .
. . , n, an 6= 0}. So S is the set of all finite ordered n-tuples
of nonnegative integers where the last coordinate is not 0. Find a
bijection from S to Z +.
Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) =...
Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q
Prove: Let n ∈ N, a ∈ Z, and gcd(a,n) = 1. For i,j ∈ N,...
Prove: Let n ∈ N, a ∈ Z, and gcd(a,n) = 1. For i,j ∈ N,
aj ≡ ai (mod n) if and only if j ≡ i (mod
ordn(a)). Where ordn(a) represents the order
of a modulo n. Be sure to prove both the forward and backward
direction.
Let n∈N, and let a1,a2,...an∈R. Prove that
|a1+a2+...+an|<or=|a1|+|a2|+...+|an|
Let n∈N, and let a1,a2,...an∈R. Prove that
|a1+a2+...+an|<or=|a1|+|a2|+...+|an|