# Tagged Questions

For questions about the formulation of a proof, not about the mathematics behind it.

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### Proof assistance for abstract algebra [duplicate]

Suppose that $F$ is a subfield of a field $E$ and let $a \in E$. Define a map $\theta :F[x] \rightarrow E$ by $\theta(f(x)) = f(a)$ for $f(x) \in F[x]$. (1) Prove that $\theta$ is a ring homomorphism ...
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### Uniqueness of sum and multiplication of numbers

so I was writing a program that took two strings and said if they were anagrams or not, and I had this idea of making each character into a number, adding them all and checking if the result was the ...
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### Show that $\frac{\alpha+y}{\alpha+n+\beta}\in (\frac{\alpha}{\alpha+\beta};\frac{y}{n})$

Suppose you assign a $Beta(\alpha,\beta)$ prior distribution for $\theta$, and the you observed $y$ heads out of $n$ spins. Show algebraically that your posterior mean of $\theta$ always lies ...
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### Proof for statement: It's impossible to find a pair of consecutive natural numbers whom digit sums would divide without reminder by 3

I am searching for a mathematical proof of this statement: It's impossible to find a pair of consecutive natural numbers whom digit sums would divide without reminder by 3. I have tried: To make a ...
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### What kind of proof is this?

Let's say that we want to prove that object A is blue. Is the following reasoning true? First assume that $A$ is indeed blue. Then, use other axioms to show that depending on a control parameter $p$,...
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### $\max\{c^Tx:Ax\le b,x\ge 0\}=+\infty$ iif it exists $j\in\{1,…,n\}$ such that $\max \{x_j:Ax\le b, x\ge 0\}=+\infty$

Show a vector $\vec c$ exists such that $\max\{c^Tx:Ax\le b,x\ge 0\}=+\infty$ if and only if it exists $j\in\{1,...,n\}$ such that $\max \{x_j:Ax\le b, x\ge 0\}=+\infty$ I'm only asking for a hint ...
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### Proving a basis and dimension

Hi I'm currently looking at proofs in linear algebra and came across this one and I'm compketely baffled Suppose $C_{ij}$ is the $2\times3$ matrix with $1$ in the $i,j^{th}$ entry and zero ...
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### How to use monotone convergence theorem to show that $\int \sum |f_n| = \sum \int |f_n|$

In https://math.la.asu.edu/~quigg/teach/courses/473/2009/lectures/11integral.pdf, pg 4 The monotone convergence theorem was used to state $$\int \sum |f_n| = \sum \int |f_n|$$ without proof. Can ...
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### How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
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### Moments of a continuous random variable (Exercise 4.3.3 from Grimmett and Stirzaker)

Let $X$ be a non-negative random variable with density function $f$. Show that $$E(X^r) = \int_0^\infty r x^{r-1} P(X > r)\,dx.$$ I tried using integration by parts to obtain \begin{align} \...
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### Suppose G and G' are isomorphic and G has m vertices of degree k. Prove that G' has m vertices of degree k

Suppose G and G' are isomorphic and G has m vertices of degree k. Prove that G' has m vertices of degree k. I don't know how to start this one
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### Need help understanding algebra steps taken in proof of why an even minus an odd is odd

I don't understand the algebra used in the below example proof from my textbook. Where does the + 1 come from? Is it okay to just add 1 anywhere you want? Or is there some rule here or reason you ...
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### Is my proof correct? If $f$ has a finite number of discontinuities on $[a, b]$, then it is integrable on $[a, b]$

Question: Suppose a function $f(x)$ over the interval $[a, b]$ is bounded and has only a finite number of discontinuous points on $[a, b]$. I intend to prove that it must be integrable on $[a, b]$. ...
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### Proof of Interceting Lines

I have this practice problem from a final exam study guide. Let $f,g$ be continuous on $[a,b]$ and $f(a)>g(a)$ but $g(b)>f(b)$. Prove that $\exists c \in [a,b]$ such that $f(c)=g(c)$. My ...
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### Simple Vacuous Proof, Correct Approach?

I am doing some practice exercises as I am starting out on proofs but I noticed that though I am getting the correct approach between vacuous and trivial proofs, I am not doing it in the same format ...
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### How does one consider what a graph looks like in a mathematical proof

Mostly I am wondering for example what it would be like to prove that a linear graph (negative slope) shifted right would look the same as one shifted up. Can you consider how a graph looks when ...
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### When is it appropriate to write “Then it follows”

I am reading a proof, and before the proof fully finishes, the author writes "Then it follows [the statement we are trying to prove] is true" I have been spending the last three hours justifying the ...
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### How to verify this relationship between area under the graph and the preimage?

Define $h : \mathbb{R} \to [0, \infty)$, Let $H = \{(x,y)| 0 \leq y \leq h(x)\}$ be the area under the graph (including the boundary) I wish to show the following is true: H = \bigcup_{c&...
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### Why is this proof that a circular cone is not a surface not rigorous?

In example $4.1.5$, page $73$ of Pressley's Elementary Differential Geometry, a "heuristic" argument is given to prove that the circular cone with vertex the origin and angle $\pi/4$, is not a surface....
### Prove that multiplication by an integer $a$ that is relatively prime to $n$ defines a bijection from $\mathbb{Z}_n-\{0\}$ to itself
If gcd$(a,n)=1$, then multiplication by $a$ defines a bijection from $\mathbb{Z}_n-\{0\}$ to itself. My working: If $n=p$ a prime, then we can use the Fermat's Little Theorem. If $n$ is not prime in ...