Suppose that D and E are sets, and D ⊆ E. Let A = P(E). Recall...

Suppose that D and E are sets, and D ⊆ E. Let A = P(E). Recall that P(E) denotes the set of all subsets of E. Define a relation R on A by

R = {(X, Y) ∈ A × A: [(X − Y) ∪ (Y − X)] ⊆ D}. So, XRY if and only if [(X−Y) ∪ (Y −X)] ⊆ D.

Prove that R is an equivalence relation on A.

Expert Solution

Colby Messinger answered 1 year ago

Let D be a division ring, and let M be a right D-module. Recall that a...

Let D be a division ring, and let M be a right D-module. Recall that a subset S ⊂ M is linearly independent (with respect to D) if for any finite subset T ⊂ S, and elements at ∈ D for t ∈ T, if sum of tat = 0, then all the at = 0. (a) If S ⊂ M is linearly independent, show that there exists a maximal linearly independent subset U of M that contains S, and...

Q2. Let (E, d) be a metric space, and let x ∈ E. We say that...

Q2. Let (E, d) be a metric space, and let x ∈ E. We say that x is isolated if the set {x} is open in E. (a) Suppose that there exists r > 0 such that Br(x) contains only finitely many points. Prove that x is isolated. (b) Let E be any set, and define a metric d on E by setting d(x, y) = 0 if x = y, and d(x, y) = 1 if x and y...

Let p be a prime and d a divisor of p-1. show that the d th...

Let p be a prime and d a divisor of p-1. show that the d th powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6

Number Theory: Let p be an odd number. Recall that a primitive root, mod p, is...

Number Theory: Let p be an odd number. Recall that a primitive root, mod p, is an integer g such that gp-1 = 1 mod p, and no smaller power of g is congruent to 1 mod p. Some results in this chapter can be proved via the existence of a primitive root(Theorem 6.26) (c) Given a primitive root g, and an integer a such that a is not congruent to 0 mod p, prove that a is a square...

Let the demand and supply functions for a commodity be Q =D(P) ∂D < 0 ∂P...

Let the demand and supply functions for a commodity be Q =D(P) ∂D < 0 ∂P Q=S(P,t) ∂S>0, ∂S<0 ∂P ∂t where t is the tax rate on the commodity. (a) What are the endogenous and exogenous variables? (b) Derive the total differential of each equation. (c) Use Cramer’s rule to compute dQ/dt and dP/dt. (d) Determine the sign of dQ/dt and dP/dt. (e) Use the Q − P diagram to explain your results. Find the Taylor series with n...

Let the demand and supply functions for a commodity be Q =D(P) ∂D < 0 ∂P...

Let the demand and supply functions for a commodity be Q =D(P) ∂D < 0 ∂P Q=S(P,t) ∂S>0, ∂S<0 ∂P ∂t where t is the tax rate on the commodity. (a) What are the endogenous and exogenous variables? (b) Derive the total differential of each equation. (c) Use Cramer’s rule to compute dQ/dt and dP/dt. (d) Determine the sign of dQ/dt and dP/dt. (e) Use the Q − P diagram to explain your results. Find the Taylor series with n...

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is...

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset of R^k...

7. Consider the following scenario: • Let P(C) = 0.2 • Let P(D) = 0.3 •...

7. Consider the following scenario: • Let P(C) = 0.2 • Let P(D) = 0.3 • Let P(C | D) = 0.4 Part (a) Find P(C AND D). Part (b) Are C and D mutually exclusive? Why or why not?C and D are not mutually exclusive because P(C) + P(D) ≠ 1 .C and D are mutually exclusive because they have different probabilities. C and D are not mutually exclusive because P(C AND D) ≠ 0 .There is not enough...

Let the market demand curve for a good be: P = 50 – Q/10. a. Recall...

Let the market demand curve for a good be: P = 50 – Q/10. a. Recall that if the market demand curve is linear, the marginal revenue curve is also linear, with the same price-axis intercept and a slope equal to twice the slope of the demand curve. Write out the marginal revenue curve for this market demand curve. b. The elasticity of demand can be written as (1/slope)*(P/Q). Find the elasticity of demand for this market demand curve at...

let E be a finite extension of a ﬁeld F of prime characteristic p, and let...

let E be a finite extension of a ﬁeld F of prime characteristic p, and let K = F(Ep) be the subﬁeld of E obtained from F by adjoining the pth powers of all elements of E. Show that F(Ep) consists of all ﬁnite linear combinations of elements in Ep with coeﬃcients in F.

Question

Suppose that D and E are sets, and D ⊆ E. Let A = P(E). Recall...

Solutions

Expert Solution

Related Solutions

Let D be a division ring, and let M be a right D-module. Recall that a...

Q2. Let (E, d) be a metric space, and let x ∈ E. We say that...

Let p be a prime and d a divisor of p-1. show that the d th...

Number Theory: Let p be an odd number. Recall that a primitive root, mod p, is...

Let the demand and supply functions for a commodity be Q =D(P) ∂D < 0 ∂P...

Let the demand and supply functions for a commodity be Q =D(P) ∂D < 0 ∂P...

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is...

7. Consider the following scenario: • Let P(C) = 0.2 • Let P(D) = 0.3 •...

Let the market demand curve for a good be: P = 50 – Q/10. a. Recall...

let E be a finite extension of a ﬁeld F of prime characteristic p, and let...