(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 +.
Prove Theorem 29.10. Let n ∈ Z+. If Ai is countable for all i =
1,2,...,n,...
Prove Theorem 29.10. Let n ∈ Z+. If Ai is countable for all i =
1,2,...,n, then A1 ×A2 ×···×An is countable.
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 ·...
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
10. Let P(k) be the following statement: ”Let a1, a2, . . . , ak
be...
10. Let P(k) be the following statement: ”Let a1, a2, . . . , ak
be integers and p be a prime. If p|(a1 · a2 · a3 · · · ak), then
p|ai for some i with 1 ≤ i ≤ k.” Prove that P(k) holds for all
positive integers k
Let A = (A1, A2, A3,.....Ai) be defined as a sequence containing
positive and negative integer...
Let A = (A1, A2, A3,.....Ai) be defined as a sequence containing
positive and negative integer numbers.
A substring is defined as (An, An+1,.....Am) where 1 <= n
< m <= i.
Now, the weight of the substring is the sum of all its
elements.
Showing your algorithms and proper working:
1) Does there exist a substring with no weight or zero
weight?
2) Please list the substring which contains the maximum weight
found in the sequence.
1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . ....
1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . . .
, ak) = 1, i.e., the largest
positive integer dividing all of a1, . . . , ak is 1. Prove that
the equation
a1u1 + a2u2 + · · · + akuk = 1
has a solution in integers u1, u2, . . . , uk. (Hint. Repeatedly
apply the extended Euclidean
algorithm, Theorem 1.11. You may find it easier to prove...
Suppose you are given a set S = {a1,a2,···,an} of tasks, where
task ai requires pi...
Suppose you are given a set S = {a1,a2,···,an} of tasks, where
task ai requires pi units of processing time to complete, once it
has started. You have one computer on which to run these tasks, and
the computer can run only one task at a time. Let ci be the
completion time of task ai, that is, the time at which task ai
completes processing. Your goal is to minimize the average
completion time, that is, to minimize n1...