Recall that in class we defined, for any set A, P(A) := {B : B ⊆...

Recall that in class we defined, for any set A, P(A) := {B : B ⊆ A}, which we called the power set of A.

a) (10) Show that, if a set A has n elements (where n ∈ {0, 1, 2, 3, 4, . . .}), P(A) has exactly 2 n elements.

b) (10) Use #4(any infinite set S has a countably infinite subset) to show that, if A is an infinite set, P(A) is uncountable.

Expert Solution

Colby Messinger answered 3 years ago

Recall that a set B is dense in R if an element of B can be...

Recall that a set B is dense in R if an element of B can be found between any two real numbers a < b. Take p∈Z and q∈N in every case. It is given that the set of all rational numbers p/q with 10|p| ≥ q is not dense in R. Explain, using plain words (without a rigorous proof), why this is. That is, present a general argument in plain words. Does this set violate the Archimedean Property? If...

if A union B = S, A intersect B = empty set, P(A) = x, P(B)...

if A union B = S, A intersect B = empty set, P(A) = x, P(B) = y, and 3x-y = 1/2, find x and y.

Let ∼ be the relation on P(Z) defined by A ∼ B if and only if...

Let ∼ be the relation on P(Z) defined by A ∼ B if and only if there is a bijection f : A → B. (a) Prove or disprove: ∼ is reflexive. (b) Prove or disprove: ∼ is irreflexive. (c) Prove or disprove: ∼ is symmetric. (d) Prove or disprove: ∼ is antisymmetric. (e) Prove or disprove: ∼ is transitive. (f) Is ∼ an equivalence relation? A partial order?

Use the set definitions A = {a, b} and B = {b, c} to express the elements in the power set P(A ∪ B).

Fill in Multiple BlanksUse the set definitions A = {a, b} and B = {b, c} to express the elements in the power set P(A ∪ B).Use {} for the empty setDo not add any spaces in your answer. Example {f} not { f } {e,f,g} not {e, f, g}{[1],[2], [3], [4], {a,b}, {b,c}, [5], [6]}

True or False? For any two events A and B, P(A∩ B) ≥ 1 − P(A∪...

True or False? For any two events A and B, P(A∩ B) ≥ 1 − P(A∪ B) True or False? For any two independent events A and B, P(A| B) = P(A| Bc ) Compute B(6, .2, 2)

In class we talked about the effects of cigarette smoking on birthweight. We defined a model...

In class we talked about the effects of cigarette smoking on birthweight. We defined a model E ( O | C = c ) = (β0)^1 + ((β1)^1 x c) where O is birthweight in ounces and C is cigarettes smoked per day. Suppose instead we used the model E(L|P =p)=(β0)^2 +((β1)^2 x p) where L is birthweight in pounds and P is packs of cigarettes smoked per day (there are 20 cigarettes in a pack). What is the relationship...

In class we considered the solution to a duopoly model with a linear demand curve P=a-b(q1+q2)...

In class we considered the solution to a duopoly model with a linear demand curve P=a-b(q1+q2) and constant marginal cost c. Consider now a market with three firms instead of two. In this case the demand curve becomes P=a-b(q1+q2+q3). Assume all three firms have the same (constant) marginal cost c. (a) State the profit maximization problem for each firm and find the corresponding first order condition for each. (b) Write the 3 first order conditions in (a) as a system...

we have defined open sets in R: for any a ∈ R, there is sigma >...

we have defined open sets in R: for any a ∈ R, there is sigma > 0 such that (a − sigma, a + sigma) ⊆ A. (i) Let A and B be two open sets in R. Show that A ∩ B is open. (ii) Let {Aα}α∈I be a family of open sets in R. Show that ∪(α∈I)Aα is open. Hint: Follow the definition of open sets. Please be specific and rigorous! Thanks!

Recall the Blue Ridge Hot Tubs problem we discussed in class. Let x1 = no. of...

Recall the Blue Ridge Hot Tubs problem we discussed in class. Let x1 = no. of Aqua-Spas produced, x2 = no. of Hydro-Luxes produced, and x3 = no. of Typhoon-Lagoons produced. Suppose we change some of the original conditions in the problem, and solve it using Solver. Use the sensitivity report given to answer the questions below: Sensitivity Report Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease # Aqua-Spa 110 0 350 80 30 # Hydro-Lux 0...

f: R[x] to R is the map defined as f(p(x))=p(2) for any polynomial p(x) in R[x]....

f: R[x] to R is the map defined as f(p(x))=p(2) for any polynomial p(x) in R[x]. show that f is 1) a homomorphism 2) Ker(f)=(x-2)R[x] 3) prove that R[x]/Ker(f) is an isomorphism with R. (R in this case is the Reals so R[x]=a0+a1x+a1x^2...anx^n)

Question