# Tagged Questions

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

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### Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...
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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 ...
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### Correct process for proof in graph theory.

I'm working on what I'm sure is a fairly basic proof in graph theory. I must prove that $Every$ $graph$ $G$ $contains$ $a$ $path$ $with$ $\delta(G)$ $edges$. $\delta(G)$ is the minimum degree of the ...
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### Can$A \cap (B' \cap C')$ be $(A \cap B') \cap (A \cap C')$?

If I use the above statement, provided that it is right, in a question, would I have to prove it as well?
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### How to show that countable union of $F_\sigma$ is $F_\sigma$

On https://www.physicsforums.com/threads/countable-intersection-of-f-sigma-sets.666055/ Is it claimed that it is obvious that countable union of $F_\sigma$ is $F_\sigma$ Can someone elaborate why ...
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### Proof formalization help: Given a vector $u$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at 2 points.

Proof formalization help: Given a vector $u$ of Euclidean length $1$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at at least 2 points. I've thought about the ...
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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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### 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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### 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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### Papers with unorthodox writing style

I'm not sure if this is the right forum for this question, in any case probably CW is appropriate? I've been looking around the mathblogosphere for the past few weeks and ran into mathgen. It's ...
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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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### Is my proof correct? 'let a,b​​∈ Z. We write A | B if A divides B. Is the relation |, symmetric, transitive and/or reflexive?'

The relationship is not symmetrical. When a relationship is symmetrical: if xRy implies yRx for all x, y ∈ A (where A is a non-empty set, and R is a relation in A) If a, b ​​∈ Z, and as a | b means ...
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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 = ...
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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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### Show that boundary of a closed set is nowhere dense

Let $H$ be a closed set then, $Cl(H) =H$ and hence the $\partial H \subset H$. Now to show that the boundary is nowhere dense, it would suffice to show that $Int(Cl(\partial H)) =\emptyset$, i.e., ...
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### Disjunctive normal form (BOTH dnf and cnf) example help

T or ( not x and y ) or ( x and y ) In class we went over examples of how many things can be both DNF and CNF... eg. (not z) or y can be thought of as both (not z) or y .... dnf ((not z) or ...
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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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### Proving the Division Algorithm using induction

Let $n \in \mathbb{N}$. For every $m \in \mathbb{Z}$, there exist unique $q, r \in \mathbb{Z}$ such that $m = qn+r$ and $0 \le r \le n-1$. We call $q$ the quotient and $r$ the remainder when dividing ...
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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 ...
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### Barendregt's Substitution Lemma (lambda calculus)

I am struggling to put words on an idea used in Barendregt's Substitution Lemma's proof. (available here) The lemma states that: If x≠y and x not free in L and M, L are $\lambda$-terms: then ...