Events A1,A2, and A3 form a partition of sample space
S with Pr(A1)=3/7, Pr(A2)=3/7, Pr(A3)=1/7. E...
Events A1,A2, and A3 form a partition of sample space
S with Pr(A1)=3/7, Pr(A2)=3/7, Pr(A3)=1/7. E is an event in S with
Pr(E|A1)=3/5, Pr(E|A2)=2/5, and Pr(E|A3)=3/5.
What is Pr(E)?
What is Pr(A2|E)?
What is Pr(E')?
What is Pr(A2'|E')?
1. Let |a2| = |a6| and a3=1, find an
2. 4. consecutive terms whose sum =...
1. Let |a2| = |a6| and a3=1, find an
2. 4. consecutive terms whose sum = 40
a2 * a3 = a1*a4 +8, find a1,a2,a3,a4
A'1(x)=2A1(x)-A2(x)-A3(x)
A'2(x)=-A1(x)+2A2(x)-A3(x)
A'3(x)=-A1(x)-A2(x)+2A3(x)
with A1(0) = 0, A2(0) = 1, and A3(0) = 5 being initial...
A'1(x)=2A1(x)-A2(x)-A3(x)
A'2(x)=-A1(x)+2A2(x)-A3(x)
A'3(x)=-A1(x)-A2(x)+2A3(x)
with A1(0) = 0, A2(0) = 1, and A3(0) = 5 being initial
values
solve linear differential equations
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 ·...
(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪...
(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪ ... ∪ An is countable. (Hint: Induction.)
(6) Let F be the set of all functions from R to R. Show that |F|
> 2 ℵ0 . (Hint: Find an injective function from P(R) to F.)
(7) Let X = {1, 2, 3, 4}, Y = {5, 6, 7, 8}, T = {∅, {1}, {4},
{1, 4}, {1, 2, 3, 4}}, and S =...
Given the augmented matrix, Find a linear combination of a1, a2,
and a3 to produce b....
Given the augmented matrix, Find a linear combination of a1, a2,
and a3 to produce b. Verify that this produces b.
1
0
3
10
-1
8
5
6
1
-2
1
6
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....
1. Suppose you are given 3 algorithms A1, A2 and A3 solving the
same problem. You...
1. Suppose you are given 3 algorithms A1, A2 and A3 solving the
same problem. You know that in the worst case the running times are
T1(n) = (10^4)n, T2(n) = 10n , T3(n) = (10^3)(n^3) log10 n (a) Which algorithm is the
fastest for very large inputs? Which algorithm is the slowest for
very large inputs? (Justify your answer.) (b) For which problem
sizes is A1 the best algorithm to use (out of the three)? Answer
the same question...
1) Suppose a1, a2, a3, ... is a sequence of integers such that
a1 =1/16 and...
1) Suppose a1, a2, a3, ... is a sequence of integers such that
a1 =1/16 and an = 4an−1. Guess a formula for an and prove that your
guess is correct.
2) Show that given 5 integer numbers, you can always find two of
the numbers whose difference will be a multiple of 4.
3) Four cats and five mice form a row. In how many ways can they
form the row if the mice are always together?
Please help...