# Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

10,072 questions
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### “Prove that ∂A ∪ Int(A) = Cl(A)” [duplicate]

The question comes from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa. I have very little knowledge of set theory and proofs, so I'm not sure how to prove this. As ...
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### Prove that $\partial A = \mathrm{Cl}(A) \cap \mathrm{Cl}(X − A)$

My textbook doesn't give me the fact that $\partial A = \mathrm{Cl}(A) \cap \mathrm{Cl}(X − A)$. We're asked to prove it. I'm given $\partial A = \mathrm{Cl}(A) − \mathrm{Int}(A)$. In a previous ...
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### “Prove that ∂A is closed given ∂A = Cl(A) − Int(A)”

Similar questions have been asked, but none with the given information. My textbook doesn't give me the fact that ∂A = Cl(A) ∩ (Cl(X – A)). If that were the case, I could just state that definition ...
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### Types of Topologies: show how they compare to each other (finer, strictly finer, coarser…)

I have very little knowledge of set theory and proof writing. This is a problem from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa. DEFINITION Let $X$ be a set and ...
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### Showing this Galois group is isomorphic to $\Bbb Z_4$ by examining the nature of its automorphisms.

Say we have the polynomial $x^6-1$, I want to find the Galois group of this over $\Bbb Q$. $x^6-1=(x-1)(x+1)(x^2-x+1)(x^2+x+1)$ And using the quadratic formula on the two quadratic polynomials we ...
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### Proof by contradiction for search problem I believe is NP-hard

Intro Given that you solve the General Sudoku Problem(any puzzle validly solved). Is there an algorithm which has a hard-coded permutation $P$ and takes as input a valid 1st row of a Sudoku grid and ...
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### “Show that a single-point set {n} is closed in the digital line topology if and only if n is even.”

I have very little knowledge of set theory and proof writing. This is a problem from Introduction to Topology: Pure and Applied by Colin Adams and Robert Franzosa. DEFINITION: Digital line ...
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### Existence and uniquenees to the equation

Prove that for every real number $x$, if $x \neq 0$ then there is a unique real number $y$ such that for every number $z$, $zy = \frac{z}{x}$. First I show that there is such that $y$, and second I ...
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### Continuity on a discontinuous interval

I got a question which asked me to prove Intermediate Value Theorem for f:X->X where X = [0,1] U [2,3] and f is continuous I assumed that the function is either in [0,1] or [2,3] and so for the ...
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### Prove that the positive integers $n$ and that cannot be written as a sum of $r$ consecutive positive integers are of the form $n=2^l$ for some $l>=0$.

Prove that the positive integers $n$ and that cannot be written as a sum of $r$ consecutive positive integers are of the form $n=2^l$ for some $l>=0$. Similar questions have been asked before but ...
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### Bellman-Ford: Shortest path with $k$-edges

Let $G=(V,E)$ be a directed graph with weights $c:E\rightarrow\mathbb{R}$, $s,t\in V$ two nodes ($s\neq t$) and $k\geq 1$. How can I use the Bellman-Ford Algorithm to determine the shortest path ...
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### Lawvere - Conceptual Mathematics - Criterion for decomposition of a map into two pieces

This is a problem from Lawvere's Conceptual Mathematics in a section talking about when we can find solutions to 'choice'/lifting diagrams. I have maps: $h\colon A \to C$, $g\colon B \to C$ with the ...
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### The proof that the Mersenne number $M_{19}$ is prime.

Here is the hint to the proof given in the book: Using the following 2 theorems: 1-If $p$ is an odd prime, then any prime divisor of $M_{p}$ is of the form $2kp +1$. 2-If $p$ is an odd prime, then ...
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### If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$

If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\displaystyle \sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$. My attempt was to use firstly AM-GM in the denominator, like $a^3+5 \geq 3a+3$ and the ...
Let $f:A\rightarrow\mathbb{R},g:B\rightarrow\mathbb{R},g \circ f:C\rightarrow\mathbb{R}$ Then we have: 1.$C\subseteq A$ 2.$f(C)\subseteq B$ Proof. $\forall c \in C,g \circ f$ is ...