5.
(a) Let σ = (1 2 3 4 5 6) in S6. Show that G = {ε, σ, σ^2, σ^3,
σ^4, σ^5} is a group using the operation of S6. Is G abelian? How
many elements τ of G satisfy τ^2 = ε? τ^3 = ε? ε is the identity
permutation.
(b) Show that (1 2) is not a product of 3-cycles. Must be
written as a proof!
(c) If a^4 = 1 and ab = b(a^2) in a...
Q: (LU decomposition) Find the LU decomposition of A = [-3 2 5
1; 12 -4 -20 -2; -6 0 15 1; -9 6 35 4]. You can use the compact
method which works within a single matrix or you can build L and U
separately. State L and U explicitly, and verify (in Matlab) that A
= L*U. Hint: Matlab's built-in lu function isn't useful, since it
pivots.
Let X = {1, 2, 3, 4, 5, 6} and let ∼ be given by {(1, 1),(2,
2),(3, 3),(4, 4),(5, 5),(6, 6),(1, 3),(1, 5),(2, 4),(3, 1),(3, 5),
(4, 2),(5, 1),(5, 3)}.
Is ∼ an equivalence relation? If yes, write down X/ ∼ .
6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f = {(1, 5),(2,
5),(3, 6),(x, y)} where x ∈ A and y ∈ B are to be determined by
you. (a) In how many ways can you pick x ∈ A and y ∈ B such that f
is not a function? (b) In how many ways can you pick x ∈ A and y ∈
B such that f : A → B...
Let S = {1, 2, 3, 4, 5, 6, 7} be a sample of an experiment and
let X = {1, 4, 7}, Y = {2, 3, 5}, and Z = {1, 3, 5} be events.
Which of the following statements is correct?
a) X and S are mutually exclusive events.
b) X and Y are mutually exclusive events.
c) X, Y, and Z are mutually exclusive events.
d) Z and Y are mutually exclusive events.
e) X and Z...
Let A = {1, 2, 3, 4, 5}. Find the inverse of the following
functions f: A→ A.
? = {(1,1),(2,3),(3,2),(4,4),(5,5)
? = {(1,5),(2,4),(3,2),(4,1),(5, 4)}
? = {(2,1),(3,4),(1,3),(4,1),(5, 2)}
4) Let ? = {2, 3, 5, 7}, ? = {3, 5, 7}, ? = {1, 7}. Answer the
following questions, giving reasons for your answers.
a) Is ? ⊆ ??
b)Is ? ⊆ ??
c) Is ? ⊂ ??
d) Is ? ⊆ ??
e) Is ? ⊆ ??
5) Let ? = {1, 3, 4} and ? = {2, 3, 6}. Use set-roster notation
to write each of the following sets, and indicate the number of
elements in...
ID
X
Y
1
2
3
2
3
6
3
4
6
4
5
7
5
8
7
6
5
7
7
6
7
8
8
8
9
7
8
10
12
11
Test the significance of the correlation coefficient.
Then use math test scores (X) to predict physics test scores
(Y). Do the following:
Create a scatterplot of X and Y.
Write the regression equation and interpret the regression
coefficients (i.e., intercept and slope).
Predict the physics score for each....