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
A computer consulting firm presently has bids out on three
projects. Let Ai = {awarded project...
A computer consulting firm presently has bids out on three
projects. Let Ai = {awarded project
i}, for i = 1, 2, 3, and suppose that
P(A1) = 0.23,
P(A2) = 0.25,
P(A3) = 0.29,
P(A1 ∩ A2) =
0.09,P(A1 ∩ A3) =
0.11, P(A2 ∩ A3) =
0.07, P(A1 ∩ A2 ∩
A3) = 0.02. Use the probabilities given above
to compute the following probabilities. (Round your answers to four
decimal places.)
(a) P(A2 |
A1) =
(b) P(A2 ∩...
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 ·...
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....
A certain system can experience three different types of
defects. Let Ai (i = 1,2,3) denote...
A certain system can experience three different types of
defects. Let Ai (i = 1,2,3) denote the event that the system has a
defect of type i.
Suppose that P(A1) = 0.25, P(A2) = 0.29, P(A3) = 0.33,
P(A1 ∪ A2) = 0.5, P(A1 ∪ A3) = 0.53, P(A2 ∪ A3) = 0.54,
P(A1 ∩ A2 ∩ A3) = 0.02
(a) Find the probability that the system has exactly 2 of the 3
types of defects.
(b) Find the probability...