measure theory – Question about a proposition in Munkres’s Analysis on Manifolds

I am reading through Munkres’s Analysis on Manifolds, and I get stuck in a proof of the lemma 18.1, that is stated as following:

Lema 18.1 Let $$A$$ be open in $$mathbb{R}^n$$; let $$g:Ato mathbb{R}^n$$ be a function of class $$C^1$$. If the subset $$E$$ of $$A$$ has measure zero in $$mathbb{R}^n$$, then $$g(E)$$ has measure zero in $$mathbb{R}^n$$.

He made out its proof in three steps. The first and second step are mentioned in the third, where he actually prove the theorem. Let me add some pictures of the third step.

(If you need pictures of the other two steps in order to solve the question above, let me know, please)

Note: A $$delta$$-neighborhood of a set $$X$$ is the union of all open cubes (in this case) with width $$delta>0$$ and centered at $$xin X.$$ The theorem 4.6 in that book states that every compact set $$K$$ that is contained in an open set $$Usubset mathbb{R}^n$$ has a $$delta$$-neighborhood contained in $$U$$.

So, the problem is here: When he covers the set $$E_k$$ by countably many cubes $$D_i$$ with certain properties, he asserts: Because $$D_i$$ has width less than $$delta$$, it is contained in $$C_{k+1}$$.

Why this is true? I mean, if each cube $$D_i$$ is centered at some point lying at $$C_k$$ it is clearly true, but we don’t know if this happens. I tried to give a proof that we can assume that each $$D_i$$ can be choosen in a way that is centered in $$C_k$$ but I couldn’t prove that.

Can you help me to justify that assertion on the book? Thanks in advance.

proof writing – Can we use Proposition for making a proposition?

There are many discussions on the difference of proposition/theorem/lemma …. e.g., https://mathoverflow.net/questions/18352/theorem-versus-proposition

Definition: Theorem, Lemma, Proposition, Conjecture and Principle etc.

Usually, I use proposition to make a statement either true or false.

But my question here is that can we use Proposition literally? In English, proposition means to make a claim/assertion/statement, or suggest something or share an opinion.

Suppose that someone found out a scheme for approximating some function $$f(x)$$. Can he use

Proposition 1. Function $$f(x)$$ in Equation ** may be approximated by using the following equations …

? In the above example, he did not prove anything. He simply did some calculus and suggest that it can be used to approximate something. Can proposition be used in this way (to make a “suggestion”)? If yes, do we have any precedent examples in the literature?

The proposition which characterizes a subset of \$mathbb{R^n}\$ using a closed set in measure theory.

This is one of the important propositions in the Lebesgue measure theory,

Proposition

Let $$E$$ be a subset of $$mathbb{R^n}$$.

Then, for all $$epsilon >0,$$ there exists an open set $$G$$ such that $$Esubset G$$ and $$m^*(G)leqq m^*(E)+epsilon.$$

I think this proposition is useful.

This proposition characterizes a subset of $$mathbb{R^n}$$ using an open set and outer measure.

I’m searching for the proposition which characterizes a subset of $$mathbb{R^n}$$ using a closed set and outer measure.

If you know such proposition, I’d like you to tell me it.

Proposition 7.13 of Hartshorne algebraic geometry(Blowing up)

Proposition is as follows-
Let $$X$$ be a noetherian scheme, $$I$$ a coherent sheaf of ideals and let $$pi$$ : $$bar{X}$$ $$rightarrow$$ $$X$$ be the blowing up of $$I$$ . Then the inverse image ideal sheaf is an invertible sheaf on $$bar{X}$$ .

Please explain me the proof . It says that inverse image ideal sheaf comes out to be $$O$$(1) on $$bar{X}$$ and now we know that it is invertible sheaf hence the result follows.

vector spaces – Some questions about a proposition that leads to definition of affine morphism

The question is about a proposition that leads to definition of affine morphism, in section 2.3 of “Geometry I” by Marcel Berger et al. To be searchable, I type the proposition and related text as follows:

Let $$(X,vec X, Theta)$$ and $$(X’,vec {X’}, Theta’)$$ be two affine spaces (over the same field), and $$f:Xto X’$$ a map (in set-theoretic sense). The following conditions are equivalent:

i) $$fin L(X_a; X’_{f(a)})$$ for some $$ain X$$;

ii) $$fin L(X_a; X’_{f(a)})$$ for all $$ain X$$;

There are more to come in the proposition, but I wish to prove i)$$Rightarrow$$ii) first, with the hope that this proof can help me deal with the remaining. If the notations are not familiar to you, in affine space $$(X,vec X, Theta)$$, $$X$$ is the affine space, $$vec X$$ is the underlying vector space, $$Theta$$ is the map from $$Xtimes X$$ to $$vec X$$ decided by simple transitivity. $$L(E;F)$$ is the set of all linear transformations from vector space $$E$$ to $$F$$. $$X_a$$ is the vectorialization of $$X$$ at $$a$$ (cf 2.1.9 of this text).

The first question is that I don’t know if my understanding of vectorialization is correct. When writing $$X_a$$, does it mean the vector space structure is given, though I don’t know what the concrete vector addition and scale multiplication is? If so, does the condition i) mean that, for some $$ain X$$, $$X$$ is endowed with a vector space structure which is isomorphic with $$vec X$$ under the map $$Theta_a$$, while at the same time, $$X’$$ is endowed with a vector space structure which is isomorphic with $$vec{X’}$$ under the map $$Theta’_{f(a)}$$?

The second question is about the proof. To establish ii) under arbitrary $$bin X$$, I think I have to construct a vector space structure on $$X$$ so that $$X$$ is vectorializable at $$b$$, as well as the same thing for $$X’$$ at $$f(b)$$. But how to do it? I just have no idea what first step to take from the given vectorialization. After that, I need to prove $$f$$ is a linear transformation between $$X_b$$ and $$X’_{f(b)}$$. But before I can endow the two affine spaces with a vector space structure, I don’t know how to proceed.

I hope I have formulated my question clearly. Please let me know if there is any clarification needed. Thank you in advance for you help with the proof.

linear algebra – Understanding 0-dim. case in proposition 15.3 in “Introduction to smooth manifolds” by Lee

I am trying to understand the proof of the following proposition. It is taken from Lee’s book “Introduction to smooth manifolds”:

I have trouble understanding the 0-dim. case. In partucluar why $$omega>0$$ implies $$mathcal{O}_{omega}$$ is +1 and the other case as well.

Since it says this case is immediate I suppose I am missing something obvious, but unfortunately I am completely stuck. I am also not sure what “consistently oriented” means in the case of a 0-dim. vector space. There is only a definition for the case $$ngeq 1$$. In Jänich’s book “Vector Analysis” I found something that would make sense to me regarding the definion of orientaion in the 0-dim. case. It is the following:

Then $$mathcal{O}_{omega}=[emptyset]$$, since the empty set is the only ordered basis in $$V=0$$?

Thank you very much in advance!

Translate statement into proposition logic

I have bit confusion to change this language into mathematical logic.
Given statement is:-

If you study, you will get good marks, if you don not study; you will enjoy.
Therefore either you will study or you will enjoy.

discrete mathematics – How to prove a proposition related to sets operation

I am asked to prove or disprove the following proposition:
$$X cap Y = X implies X cup Y = Y$$

I feel this is true, because this means that X is a subset of Y, and as a result, when we do union with these two, it should be the entire set.

However, I have a hard time constructing a formal proof to prove this. Not sure where to start from.

logic – proposition formula – Mathematics Stack Exchange

I meet a case that for four letters P, Q, R, S, only when P=0 AND Q=0 AND R=1 AND S=1 or P=1 AND Q=1 AND R=0 AND S=0, the output will be true, all others are false. The thing here is to express this formula using each letter only once. Namely, using P()Q()R()S to express this formula. Is it possible?

Why is the Value Proposition canvas represented with a square and a circle?

I was wondering why the value proposition model is represented with a square for the value proposition and a circle for the customer profile.

Is there an explanation for it? or is it just something visual?

I don’t know why it just makes more sense to me to represent it with two triangles each divided in three sections…just a thought