Question

Let X be a set and let F1,F2 ⊆ P(X) be two σ-algebras on X. Let G := {A ∩ B | A ∈ F1, B ∈ F2}. Prove the following statements:

(1) G is closed under finite intersections.

(2) σ(G) = σ(F1 ∪ F2).

Answer #1

Let f1, f2, f3: [a,b] -->R be nonnegative concave functions
such that f1(a) = f2(a) = f3(a) = f1(b) = f2(b) = f3(b) = 0.
Suppose that max(f1) <= max(f2) <= max(f3).
Prove that: max(f1) + max(f2) <= max(f1+f2+f3)

Consider the following predicate formulas.
F1: ∀x ( P(x) → Q(x) )
F2: ∀x P(x) → Q(x)
F3: ∃x ( P(x) → Q(x) )
F4: ∃x P(x) → Q(x)
For each of the following questions, answer Yes or No &
Justify briefly
. (a) Does F1 logically imply F2?
(b) Does F1 logically imply F3?
(c) Does F1 logically imply F4?
(d) Does F2 logically imply F1?

Let
X be finite set . Let R be the relation on P(X). A,B∈P(X) A R B Iff
|A|＝|B| prove R is an equivalence relation

Determine if the set of functions is linearly independent:
1. f1(x)=cos2x, f2(x)=1, f3(x)=cos^2 x
2. f1(x)=e^ x, f2(x)=e^-x, f3(x)=senhx

1. There are two auto producers in Karmania, F1 and F2. The cars
they produce are essentially identical. The marker inverse demand
curve is given by p = a - bQ, where p is the price (in thousands of
dollars); Q market output (in thousands of units); and a and b are
parameters. It is estimated that a = 25 and b = 0.1. Both F1 and F2
have a marginal cost of 10 thousand dollars per car. Competition in...

Let X be a set and A a σ-algebra of subsets of X.
(a) A function f : X → R is measurable if the set {x ∈ X : f(x)
> λ} belongs to A for every real number λ. Show that this holds
if and only if the set {x ∈ X : f(x) ≥ λ} belongs to A for every λ
∈ R. (b) Let f : X → R be a function.
(i) Show that if...

Let S be a finite set and let P(S) denote the set of all subsets
of S. Define a relation on P(S) by declaring that two subsets A and
B are related if A and B have the same number of elements.
(a) Prove that this is an equivalence relation.
b) Determine the equivalence classes.
c) Determine the number of elements in each equivalence
class.

Two forces F1 and
F2 act on an object. Each
force is specified as an ordered pair: its magnitude in Newtons and
an angle measured in degrees counter-clockwise from the positive
x-axis.
Let F1 = (27.0 N, ∠115°
)and F2 = (83.0 N, ∠240°
).
Calculate the magnitude the vector
F1 -
F2. Include three significant
figures in your answer. (If the last digit is a zero, Canvas will
not display it but that's OK.)
Please HELP!

Prove that (Orbit-Stabilizer Theorem). Let G act on a finite set
X and fix an x ∈ X.
Then |Orb(x)| = [G : Gx] (the index of Gx).

Let X be a non-empty finite set with |X| = n. Prove that the
number of surjections from X to Y = {1, 2} is (2)^n− 2.

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