Question

In: Advanced Math

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 so, how? (PLEASE DON'T REPEAT THE ANSWERS TO THIS QUESTION ALREADY POSTED ON CHEGG)

Expert Solution

Sorry, I don't know about the previous answers. I hope this one is clear to you.

Let

.

If is dense in then by defenition, for ANY real number a and b, a<b, we can find an element in .

Now, we have to prove that is not dense in .

For that it is enough to prove that

.

----------------------------(i)

So let me choose

Then clearly

which means if we choose ,

we cannot find an element of between .

ARCHIMEDEAN PROPERTY

For any and , there is an so that which is equivalent to .

doesnot satisfies Archimedean property.

Because,

For any natural number , there exists a positive real number

where such that , which is a contradiction to Archimedean property.

Colby Messinger answered 2 years ago

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.

Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be...

Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be the usual “less than or equal to” total order on the set A=(−2,0)∪(0,2) Consider the subset B={−1/n | n∈N, n≥1}⊆A. Determine an upper bound for B in A. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound. Note: least upper bound is an upper...

. Recall from the previous page that for each pair a, b with a ∈ R...

. Recall from the previous page that for each pair a, b with a ∈ R − {0} and b ∈ R, we have a bijection fa,b : R → R where fa,b(x) = ax + b for each x ∈ R. (b) Let F = {fa,b | a ∈ R − {0}, b ∈ R}. Prove that the set F with the operation of composition of functions is a non-abelian group. You may assume that function composition is associative

Prove that {??+?:?,?∈?} is dense in ? if and only if r is an irrational number.

4. Show that the set A = {fm,b : R → R | m does not...

4. Show that the set A = {fm,b : R → R | m does not equal 0 and fm,b(x) = mx + b, m, b ∈ R} forms a group under composition of functions. (The set A is called the set of affine functions from R to R.)

In R, Use the anorexia data set for questions 8-15 Recall there are three distinct treatments....

In R, Use the anorexia data set for questions 8-15 Recall there are three distinct treatments. You may be suspicious that there is a significant difference in the mean weights assigned to the different treatments. It is not a good experimental design if one of the treatments has subjects who are starting in a better position than the subjects of other treatments. To test if this is a well-designed experiment we perform an ANOVA analysis .The null hypothesis is the...

Let the schema R = (A,B,C) and the set F = {A → B,C → B}...

Let the schema R = (A,B,C) and the set F = {A → B,C → B} of FDs be given. Is R in 3NF? Why or why not?

On the set S of all real numbers, define a relation R = {(a, b):a ≤ b}. Show that R is transitive.

Question

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

Solutions

Expert Solution

Related Solutions

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

Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be...

. Recall from the previous page that for each pair a, b with a ∈ R...

Prove that {??+?:?,?∈?} is dense in ? if and only if r is an irrational number.

Prove that {??+?:?,?∈?} is dense in ? if and only if r is an irrational number.

Prove that {??+?:?,?∈?} is dense in ? if and only if r is an irrational number.

4. Show that the set A = {fm,b : R → R | m does not...

In R, Use the anorexia data set for questions 8-15 Recall there are three distinct treatments....

Let the schema R = (A,B,C) and the set F = {A → B,C → B}...

On the set S of all real numbers, define a relation R = {(a, b):a ≤ b}. Show that R is transitive.