For each of the following sets, determine whether they are
countable or uncountable (explain your reasoning)....
For each of the following sets, determine whether they are
countable or uncountable (explain your reasoning). For countable
sets, provide some explicit counting scheme and list the first 20
elements according to your scheme. (a) The set [0, 1]R ×
[0, 1]R = {(x, y) | x, y ∈ R, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.
(b) The set [0, 1]Q × [0, 1]Q = {(x, y) |
x, y ∈ Q, 0 ≤ x ≤...
1. Which of the following sets in (a) are groups under addition?
For each set which...
1. Which of the following sets in (a) are groups under addition?
For each set which is not a group under addition, show which group
property does not apply by counterexample.
a. N; W; Z; Q; R; E; C; P(x, 3); M(2,1,N) .
Indicate whether each statement is True or False.
Briefly justify your answers. Please answer all of...
Indicate whether each statement is True or False.
Briefly justify your answers. Please answer all of questions
briefly
(a) In a vector space, if c⊙⃗u =⃗0, then c= 0.
(b) Suppose that A and B are square matrices and that AB is a
non-zero diagonal matrix. Then A is non-singular.
(c) The set of all 3 × 3 matrices A with zero trace (T r(A) = 0)
is a vector space under the usual matrix operations of addition and
scalar...
2.For each of the following, give a concrete example. Explain in
max. 3 lines why your...
2.For each of the following, give a concrete example. Explain in
max. 3 lines why your example has the stated property.
(c) An equivalence relation on N that has exactly three
equivalence classes.
(d) An ordering relation on the set {a, b, c, d} that does not
have a maximum element.
1. [10 points] For each of the following statements, indicate
whether it is true or false. You don’t have to justify your
answers.
(i) If R is an equivalence...
Determine if each of the following statements is true or false.
If it’s true, explain why....
Determine if each of the following statements is true or false.
If it’s true, explain why. If it’s false explain why not, or simply
give an example demonstrating why it’s false
(a) If λ=0 is not an eigenvalue of A, then the columns of A fo
ma basis of R^n.
(b) If u, v ∈ R^3 are orthogonal, then the set {u, u − 3v} is
orthogonal.
(c) If S1 is an orthogonal set and S2 is an orthogonal set...
1. Which of the following set of quantum numbers (ordered n, ℓ,
mℓ, ms) are possible...
1. Which of the following set of quantum numbers (ordered n, ℓ,
mℓ, ms) are possible for an electron in an atom? Check all that
apply.
a. 3, 2, 2, -1/2
b. -1, 0, 0, -1/2
c. 3, 2, 1, -1
d. 4, 3, 4, -1/2
e. 2, 2, 2, 1/2
f. 4, 3, -2, 1/2
g. 5, 2, 1, -1/2
h. 3, 1, -2, -1/2
2. How many possible combinations are there for the values of l
and ml...
19. Which of the following sets of quantum number is not
allowed? A. n = 4,...
19. Which of the following sets of quantum number is not
allowed? A. n = 4, ℓ = 2, mℓ = -2 B. n = 5, ℓ = 0, mℓ = 0 C. n = 6,
ℓ = 3, mℓ = 1 D. n = 3, ℓ = 1, mℓ = −1 E. A–D are all allowed sets
of quantum numbers