This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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24 views

Let $F$ be a class of sets. Prove that $B - \mathop{\bigcup}_{A\in F} A = \mathop{\bigcap}_{A\in F} (B-A)$

Let $F$ be a class of sets. Prove that $B - \mathop{\bigcup}_{A\in F} A = \mathop{\bigcap}_{A\in F} (B-A)$ I've started like this: $X= B - \mathop{\bigcup}_{A\in F} A$ $Y= \mathop{\bigcap}_{A\in ...
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4answers
57 views

Prove that if $a<b$, then $-b<-a$

Prove that if $a<b$, then $-b<-a$ I'm a bit lost in this one, this is what I did: First case: $0<a<b$ $|a|<|b|$, so $|-a|<|-b|$ Since both are negative and $|-b|$ is greater ...
2
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1answer
77 views

How to remember all the proofs in mathematics

I have a problem where I forget the proof of a theorem after some time without reworking it out. However, my teacher said that he was able to prove a theorem even without reworking it out for a long ...
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1answer
41 views

Ten men selected to include a captain vs ten men selected to include at least one officer

I can't see how these are different scenarios, but the answers are different. The general scenario is that there is a company of volunteers that consists of a captain, a lieutenant, an ensign, and ...
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0answers
28 views

Help on Lagrange Error Calculation [duplicate]

Here is an example in Burden's Numerical Analysis book. My problem is in bold In example 2 we found the second Lagrange polynomial for $f(x)=1/x$ on $[2,4]$ using the nodes $x_0=2$, $x_1=2.75$, ...
2
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2answers
78 views

Confusion on the proof that there are “arbitrarily large gaps between successive primes”

I am trying to wrap my brain around a proof that proves that there are arbitrarily large gaps between successive primes. The proof is Given a natural number $N\ge2$, consider the sequence of $N$ ...
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2answers
36 views

How can the only maximal ideal of $C[x] / X^2$ be $(X)$?

In my notes I have the following example which I don't understand. Let $f$ be the canonical injection from $C$ to $C[X]/X^2$.The only maximal ideal of $C[X]/X^2$ is $(X)$ and $f^{-1}((X))$=$(0)$. ...
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1answer
68 views

Help needed to understand statements about torus

I am having trouble understanding two statements: Let $A$ be an algebraic curve in $\mathbb{P}^2$ over $\mathbb{C}.$ Consider its normalization $$\pi: \hat{A} \to A.$$ If genus $g(\hat{A})=1,$ ...
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2answers
66 views

Proof of the Intermediate Value Theorem [closed]

Theorem: Let $f$ be continuous on $[a,\,b]$ and assume $f(a)<f(b)$. Then for every $k$ such that $f(a)<k<f(b)$, there exists a $c\in[a,\,b]$ such that $f(c)=k$. proof: $f$ continuous at ...
3
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0answers
39 views

The proof of remainder theorem for polynomials: $f(\lambda)=0$ if and only if $x-\lambda$ divides $f$

From the paragraph starting with "Let us now suppose that $f(\lambda) = 0$" the author states that if $\deg(f) \leq 0 $ then $f=0$. But if $\deg(f) = 0$ then $f$ can be any constant, and not ...
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1answer
51 views

strong induction example

There is following example given in a book. I am not sure how do we conclude that $a$ is divisible by prime? See this section: Case 2 ($k + 1$ is not prime): In this case $k+1=ab$ where $a$ and ...
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2answers
46 views

Explanation of this proof of the principle of analytic continuation

I am confused about the last paragraph, He states that we choose a point $z_1$ inside the disk, and draw another disk around $z_1$ with radius $\delta_1$ Then says "f is identically zero on this ...
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1answer
23 views

Principle of analytical continuation proof confusion

Let $f$ be holomorphic on a region $D$ and assume I can find a sequence $\{w_k\}$ in $D$ s.t. $w_k \to z_0$ w/ $z_0 \in D$, $w_k \not = z_0$ & $f(w_k) = 0$. Then $f(z) = 0$ $\forall z \in D$ ...
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1answer
38 views

Fundamental Characterisation of Measureable Functions Explanation (Characterization)

I am just starting learning about measure theory (what a great way to spend Christmas...!), and I am unclear on this following claim: Characterisation of Measurable Functions. It appears to be a very ...
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3answers
74 views

Just a proof of algebra

If $a+b+c=0$, Show that $\left[\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}\right]\left[\dfrac{b-c}{a}+\dfrac{c-a}{b}+\dfrac{a-b}{c}\right]=9.$ I am struck with this problem but can't find a ...
2
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2answers
41 views

In Zagier's one-sentence proof, why is S defined to be {(x,y,z)∈ℕ^3:x^2+4yz=p,p prime}?

I've looked at a very clear explanation of Zagier's proof (specifically, it can be found here:http://danielmath.wordpress.com/2012/12/26/one-sentence-proof/) but the first step still eludes me: why is ...
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1answer
54 views

Localization of Rings: Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$

Let $R$ be a ring, $f \in R$, and $X$ a variable. Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$. I am a beginner in algebra and I am reading a textbook in commutative algebra. What I do not ...
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1answer
47 views

Explanation of Proof that field sum of more than 2 elements is 0. [duplicate]

"Suppose the field $F$ is finite. If $f\colon F\to F$ is any bijection, then we can conclude that $\sum_{x\in F}x=\sum_{x\in F}f(x)$. Let $\alpha\in F$ such that $\alpha\ne 0$. Then $x\mapsto \alpha ...
2
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0answers
48 views

Is proper morphism from affine scheme affine?

I'm reading Mumford-Oda's lecture notes http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap1-6.pdf. And they use the fact:"Let $f:U \to Y$ be a proper morphism of noetherian schemes and $U$ is ...
2
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2answers
55 views

Prove that the unity element in a subfield of a field must be the unity of the whole field

The solution I was given says: Let $F$ be a field and suppose $u^2=u$ for some nonzero $u$ in $F$. By multiplying each side by $u^{-1}$ it is clear that $0$ and $1$ are the only solutions of ...
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1answer
32 views

Explaining That there is no nontrivial ring homomorphism between Z and nZ

My instructor wrote in his notes the following example: "As groups (Z,+) and (nZ,+) are isomorphic. As rings is there any nontrivial homomorphism $\phi$: Z->nZ? The answer is no and he gives the ...
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1answer
90 views

Construction of a triple cover of $A_6$ in “Finite Simple Groups” by Wilson

I am reading The Finite Simple Groups by Robert Wilson: see page 29. I want to understand a construction of triple cover of $A_6$. On section 2.7.3., I don't understand the second paragraph, which is ...
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1answer
68 views

Satellite functors in Cartan Eilenberg

I was reading and came across this statement whose proof is said to be obvious. I however after hours still cannot figure out how to prove $S_2T(A) = S_1(S_1T(A)) = S_1T(M)$. The definitions are: ...
3
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1answer
61 views

Doubt on a paragraph regarding Lagrange's multiplier.

I've a topic in my notes "The method of Lagrange's multipliers" which is described as follows: Let $U$ be an open set in $\mathbb R^n$.Let $f\in C^1(U,\mathbb R)$ and let ...
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1answer
36 views

Question on abstract algebra about Group?

I need an explanation, why $ (\mathbb{Z}_7,\oplus _6 )$ is not a Group? As I have discovered so far. The following conditions are satisfied I) Closed! II) Associative! III) ...
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1answer
18 views

Sequence Lemma explanation

Then every neighbourhood $U$ of $x$ contains a point of $A$. So I don't see it happening unless $X$ is a metric space, but the proof is for any topological space.
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1answer
31 views

Haudorff Formula Set Theory

For every $\alpha$ and every $\beta$, $$\aleph_{\alpha+1}^{\aleph_{\beta}}=\aleph_{\alpha}^{\aleph_{\beta}} \cdot \aleph_{\alpha+1}$$ Proof: If $\beta \geq \alpha+1$, then ...
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0answers
25 views

Fichtenholz's limit proof for a sequence

I'm reading calculus written by G.M Fichtenholz. And i can't understand his proof. The proof is: $n>2$ There is a sequence $x_n = \frac{n^2 -n + 2}{3n^2+2n-4}$ $x_n - \frac13 = \frac{5n - 10}{3( ...
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1answer
22 views

Question regarding an algebraic manipulation in GFology

How does the author arrive at the last equality in the first line, i.e.$$\text{why is } [x^k]\frac{1}{1-y(1+x)} = \frac{1}{1-y}[x^k]\frac{1}{1-\left(\frac{y}{1-y}\right)x} \text{?}$$
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2answers
58 views

set theory proof explanation

The following lemma is taken from the book 'Introduction to Set Theory' by Hrbacek and Jech. chapter $6$ normal form Can anyone explain to me why the first sentence holds ( the existence of ...
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0answers
25 views

Multilinear Function proof in Spivak?

Note that $$\wedge^n (V)$$ denotes the set of all alternating multilinear functions and $\mathfrak{I}^n(R^n)$ denotes the set of all multilinear function. I don't know what the actual symbol ...
2
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1answer
31 views

Multilinear algebra and matrices

Given $\wedge^k(V)$ an alternating multilinear space and $T : V \to W$ a linear map, then we have $$v_1 \wedge \dots \wedge v_k \in\wedge^k(V).$$ Define $$\wedge^k(T)(v_1\wedge\dots\wedge v_k) = ...
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1answer
42 views

Understanding the proof that $c_0$ is a closed subspace of $\ell^\infty$

The problem is given: source Let $c_0$ be a space of real sequences $x = \{x_n\}_{n = 1}^\infty=0$ converging to $0$. Let $\ell^\infty$ be a set of real sequences $w = \{w_k\}^\infty_{k=0}$ ...
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1answer
75 views

A limit theorem in Rudin. Please elaborate?

Theorem: Let $\{ p_n \} \in X $. If $E \subset X$ and $p$ is a limit point of $E$, then there is a sequence $\{ p_n \}$ such that $p = \lim p_n $ The proof goes like this proof: For each ...
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0answers
39 views

Transfinite Induction (Proof Explanation)

The theorem above is extracted from the book 'Introduction to Set Theory' by Hrbacek and Jech. Questions: $1$)I don't understand the successor case. When $\alpha_2=\beta+1$, why suddenly $W_2$ ...
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1answer
39 views

Help understanding example in Engel's *Problem Solving Strategies*

I've spent a lot of time trying to follow the chain of reasoning, but to no avail. I lose track of how it works at the "Adding (1) and (2)" part. Could someone help me understand this, please?
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1answer
49 views

Proposition 0.16 in Hatcher's AT

In the proof of the quoted proposition, it is mentioned that $D^n \times I$ retracts onto $D^n \times \left\{0\right\} \cup \partial D^n \times I$ and an example is given in a figure with $n=2$, which ...
0
votes
2answers
77 views

$1$ is not congruent because of Fermat's Last Theorem?

I would like someone to explain something I did not understand. I was reading a page called "nuking the mosquito" where they give very complex proofs for very simple results. The proof I want to talk ...
2
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1answer
37 views

Proposition 1A.1 in Hatcher's Algebraic Topology

In the proof of the quoted Proposition, we have a connected graph $X$ and a sequence of subgraphs $X_0 \subset X_1 \subset \cdots$ such that $\cup_i X_i$ is both open and closed. Then Hatcher deduces ...
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3answers
52 views

How to determine value from willingness to pay?

I use the British pounds symbol instead of dollars because $ conflicts with Mathjax. Source: p 296, The Legal Analyst, Ward Farnsworth "... one time in a thousand we do lose the film; if you’re ...
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1answer
50 views

Proposition I.6.8 in Hartshorne

In the context of the quoted proposition arises the following question. Let $X$ be an abstract nonsingular curve (as defined in p. 42), $P \in X$ and let $\phi: X-P \rightarrow \mathbb{P}^n$ a ...
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1answer
39 views

Proposition I.6.7 in Hartshorne

Proposition (I.6.7,HAG): Every nonsingular quasi-projective curve $Y$ is isomorphic to an abstract nonsingular curve. The first paragraph of the proof establishes a bijective map $\phi: Y \rightarrow ...
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1answer
67 views

Hartshorne Theorem I.5.3

This question concerns a reduction argument that occurs in the proof of Theorem I.5.3 in Hartshorne. In particular, let $Y$ be an affine variety of dimension $r$ in $\mathbb{A}^n$. Then by (4.9) $Y$ ...
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1answer
52 views

Weibel “Introduction to homological algebra” Main Theorem 4.4.16

I can't understand the proof of Main Theorem 4.4.16 from Weibel's book "An Introduction to homological algebra". The Theorem states Let $R$ be a local noetherian commutative ring, then $R$ is ...
2
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1answer
43 views

Obscure inequality in a passage of a proof (maximal ergodic theorem)

Look at the following excerpt from the book "Einsiedler and Ward- Ergodic Theory, with a view towards Number Theory": I don't understand why the inequality in the red box is valid. Maybe the ...
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1answer
31 views

Question in Fulton and Harris regarding induced representation.

I'm confused by the following paragraph: I don't see why $g\cdot W$ depends only on the left coset $gH$. What does he mean precisely by that? Why is it true that $gh\cdot W = g\cdot(h\cdot W) = ...
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1answer
55 views

Groups of order 8 proof

I understand the solution to these questions I was just wanting to confirm that the solution to Q10 excludes the possibility that $y =x^2$ ( and hence the proof is not complete) as it uses the ...
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5answers
83 views

What is the limit of $\log_k(k^a + k^b)$ for $k \to +\infty$?

I'm not very good with analysis (I never studied it) but because of my "work" on other topics of mathematics I came to this problem. $$\lim_{k \to +\infty }\log_k(k^a + k^b)=\max(a,b)$$ I'm sure ...
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1answer
34 views

The topology of the restriction of a metric is the restriction of the topology.

I'm reading this proof of the following claim. Let $(X,d)$ be a metric space and $(Y,d')$ a subspace of $X$. If $(X,T)$ is the topology induced by $d$ and $(Y,T')$ the typology induced by $d'$, ...
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1answer
171 views

Hartshorne Theorem 8.17

I can't understand the proof of theorem 8.17 from Hartshorne's "Algebraic Geometry". Namely, he says that we have an exact sequence $$ 0 \to \mathcal J'/\mathcal J'^2 \to \Omega_{X/k} \otimes ...