## elementary number theory – Prove the set of even integers is well-ordered using the well-ordering principle?

I’m a beginner. According to my professor, I must write out every step as clearly as possible.

Question: Prove the set of even integers is well-ordered using the well-ordering principle.

Solution (According to Bartelby.com): By definition, the set of positive even integers is the subset of positive integers. Moreover, since the even integers are well-ordered, (by the well-ordering principle), the positive even integers are well-ordered.

However, I’m unable to understand the proof. How do we know that if a set has the property then its subset has the property? (My Professor demands I rely on proofs rather than intuition).

Below is my attempt at the problem:

My attempt: By definition, the set of positive even integers is the subset of positive even integers. Hence if the set of positive even integers are well-ordered, then; the set of postive integers are well-ordered. Despite this, we must prove if positive integers are well-ordered then positive even integers are well-ordered. Therefore, we take the contrapositive.

If the positive integers are not well-ordered then the positive even integers are not well-ordered; however, the positive integers are well-ordered (by the well-ordering principle). This is a contradiction to our hypothesis where the positive integers are not well-ordered. Hence if the positive integers are well-ordered then the positive even integers are well-ordered. Therefore, the positive even integers are well-ordered.

Question: Am I correct or have I made a mistake in my steps? Try and explain step by step?

## Prove with pumping theorem that the language { $a^n b^n b^m a^m | n ≠ m$ } is not context free

I’m having a trouble proving it to be context free, with an example if I take w = $$a^k b^k b^{k+1} a^{k+1}$$

then it would be problematic if the partition of $$vxy$$ with $$|v| = |y|$$ was in the $$b^{k+1} a^{k+1}$$ part as I can’t get it to be equal with $$k$$ or make the amount of $$a$$ and $$b$$ different

It is also allowed to be proven with Ogden’s Lemma

## Prove language is not Turing-recognizable using contradiction

Show that the language L = {<M>| M is a TM and does not accept <M>} is not Turing-recognizable.

Note: Prove by contradiction. No need for reduction.

This is the problem I am trying to solve. I’m confused on how to do this without using reduction.

## sorting – How to prove that a binary heap can’t be sorted with complexity that better than O(nlogn)

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## functional analysis – How to prove that support of $u_{r}$ is compact?

Let’s $${u_r}:{mathbb{R}^k}to{mathbb{R}}$$, $$varphi in C_{c}(mathbb{R}^n)$$, $$Jf$$ ( jacobian of $$f$$, with $$f: mathbb{R}^{k}to mathbb{R}^{n}$$ Lipschitz ), and $$chi_E$$ (characteristic function on $$E subset mathbb{R}^{k}$$ where $$E$$ borel bounded subset),
$$begin{equation*} u_r(w) := chi_E(z+rw)varphiBigl(frac{f(z+ rw)-f(z)}{r}Bigr) J f(z+rw). end{equation*}$$

Statement: there are $$r_0> 0$$ and $$R> 0$$ such that $$supp(u_{r}) subset mathbb{B}(0,R)$$ for $$r in (0, r_{0})$$.

My attempt:
In fact, since $$f$$ is derivable into $$z$$ and $$Jf(z)> 0$$, there are $$s_{0}, lambda > 0$$ such that
$$begin{equation} |{f(z’)-f(z)}| geq lambda ||{z’-z}|| end{equation}$$
for every $$z ‘in mathbb{B}(z,s_{0})$$. On the other hand, if $$rho> 0$$ is such that $$supp (varphi) subset mathbb{B}(0,rho)$$, then
$$begin{equation*} |{f(z+rw) -f(z)}|leq rrho end{equation*}$$
for all $$win supp(u_{r})$$. Hence, if $$w in supp(u_{r})$$ with $$r < s_{0} / rho$$, you have $$z + rw in mathbb{B}(z,s_{0})$$, so $$r rho geq |{f (z + rw) -f (z)}| geq lambda ||{z + rw-z}|| = lambda r ||{w}||$$, then $$||{w}|| leq rho / lambda$$, which proves statement with $$r_{0}: = s_{0} / rho$$ and $$R: = rho / lambda$$.

I’m not sure what $$z + rw in mathbb{B}(z,s_{0})$$ , I really appreciate it if someone could give me an idea how to improve this argument.

Another idea that I had, was to prove that the $$supp(u_{0})$$ is compact, which I did, in order to arrive at that the $$supp(u_{r})$$ is compact, which would be another way to conclude that statement in another way. But unfortunately I could not find that relationship between the $$supp(u_{0})$$ and the $$supp(u_{r})$$I really appreciate the attention given.

## axiom of choice – Does ZF + BPI alone prove the equivalence between “Baire theorem for compact Hausdorff spaces” and “Rasiowa-Sikorski Lemma for Forcing Posets”?

Rasiowa-Sikorski Lemma (for forcing posets)is the statement: For any p.o. $$mathbb{P}$$ (i.e. $$mathbb{P}$$ is a reflexive transitive relation) and for any countable family of dense subsets of $$mathbb{P}$$ there is a generic filter which intersects all dense subsets of the countable family. It is well-known that this statement is equivalent to the Baire Category Theorem for Complete Metric Spaces – and thus it is also equivalent to the Principle of Dependent Choices.

A masters student of mine has found in the literature the following statement: “Rasiowa-Sikorski Lemma is equivalent to the Baire Category Theorem for Compact Hausdorff Spaces, modulo the Boolean Prime Ideal Theorem”. We understood this as the assertion that the theory ZF + BPI alone is able to prove the equivalence between the Baire Category Theorem for Compact Hausdorff Spaces and the Rasiowa-Sikorski Lemma.

Well, I asked my student to verify such claim, and at first glance I suggested him to follow the results 3.1 to 3.4 of Chapter II of Kunen’s book, where there are proofs for some equivalences of Martin’s Axiom at $$kappa$$, MA($$kappa$$): the idea was to discard the hypothesis “c.c.c.” and adapt the reasoning, arguing for $$kappa = omega$$. It turns out that it was not a good suggestion, because in 3.1 a kind of Downward-Lowenheim-Skolem argument is done, to show that it is equivalent to work with a restricted form of the forcing axiom, considering only partial orders of bounded cardinality. However, such argument seems to require the Axiom of Choice, or some part of it other than BPI.

Does any of you know if it is indeed possible to prove the equivalence between “Baire Category Theorem for Compact Hausdorff Spaces” and “Rasiowa-Sikorski Lemma for forcing posets” from ZF + BPI alone ? Any suggestions or references would be appreciated.

## How to prove a surjective homomorphism is an isomorphism

Where the function f is equal to the surjective homomorphism G → H

and G and H are non trivial groups

How would I go about proving that f is an isomorphism?

## cotangent bundles – How to prove a subspace of symmetric matrix space is a manifold

I am studying Differential Geometry and came across to this problem:

Prove that the set of all idempotent symmetric matrices $$n times n$$ of rank $$k< n$$ is a manifold and I need to find its tangent space. I know how to give a proof when it contains just symmetric matrices, but now it became a big problem to me.

I appreciate any tip or solution to this question.

## Prove x . y’ . z’ is y’ . z’

Please, how can I prove that x . y’ . z’ simplifies to y’ . z’?

I have tried without success. Below is the context of my question; I am taking a course on Coursera

## how to prove this calculus 2 result?

if f(x)>=0 for every a<x<inf , and the integral from (a to inf) of f(x)sin(x) exists
then the integral from (a to inf) of f(x) exists

PS: this is my first question so i don’t know how to use Mathjax if someone could edit it i would appreciate it