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Questions tagged [proof-verification]

For questions concerning a specific proof or a specific solution, asking for verification, identifying errors, suggestions for improvement, etc. (You should not use this tag if the question does not contain a proposed proof/solution.)

For questions concerning a specific proof (or a proof sketch) or a solution to some problem; asking a question with this tag indicates one would like answers to respond broadly as to the following:

  • Verification of the proof/solution;
  • Identifying errors in the proof/solution;
  • Suggestions for improving the proof/solution;
  • Alternative approaches.

Also, consider the related tags and .

22798 questions
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Prove B $\cap$ $\bigcup_{\alpha\in\Delta}A_\alpha = \bigcup_{\alpha\in\Delta}(A_\alpha\cap B)$

I am not sure if the way I tried proving this problem is correct and so I would appreciate the feedback. My proof: (i) Let x $\in B \cap\bigcup_{\alpha\in\Delta}A_\alpha $. Then x $\in B$ and $x\in \bigcup_{\alpha\in\Delta}A_\alpha$. Then X $\in B$…
Jr194
  • 109
2
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1 answer

Prove that $g(f(x))$ is injective

$F: [0,π/2]\to \mathbb{R}$ where $F(x) = \text{sin(x)} + 77$ and $G: \mathbb{R} \to \mathbb{R}$ where $G(y) = y^4$ Show that $G(F(x))$ where domain is $[0,π/2]$ and codomain is $\mathbb{R}$ is injective. I'm sorry for not writing this neatly. The…
Hazard
  • 191
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2 answers

For every natural number $x$, $\frac{1}{x} \le 1$

I am quite new to proofs, and I tried solving this proof myself. Here is what I got: let $x$ be a natural number Suppose $\frac{1}{x} > 1$, then $1 > x $ This is a contradiction because x cannot equal $0$ and $x$ must be greater than $0$.…
Jr194
  • 109
2
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1 answer

Rewrite $x \notin A$ using negation proper way?

Suppose $x \notin A$. If I were to rewrite using negation symbol $\lnot$, which one is true? $x \lnot \in A$ $x \in \lnot A$ I'm guessing the first, because the second assumes $x$ does belong somewhere else (specifically $A^c$), but this…
MinYoung Kim
  • 1,015
2
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4 answers

Avoid ignoring the solution $0$ of an equation

Suppose I am trying to solve the equation $49r^2=9r$. I would go about solving it like this: $$49r^2=9r$$ $$49r^2-9r=0$$ $$\frac{49r^2-9r}{r}=\frac{0}{r}=0$$ $$49r-9=0$$ $$49r=9$$ $$r=\frac{9}{49}$$ But the problem is, $0$ is also a solution. How…
Bill
  • 169
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2 answers

Is it true that x in base n is always larger than or equal to x in base n-1?

I'm an elementary education major, but one of the problems in my elementary mathematics class asked to compare the values of numbers in different bases, without converting to base ten. One example was whether 32 in base 10 is larger than, equal to,…
Ashley
  • 23
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1 answer

Given $B \subseteq A$ and $x \in A$, x could also be in B

This statement should be true right? If $B \subseteq A$ and $x \in A$, $\exists x(x \in B)$. I was trying to prove this statement but my previous approach was wrong. My previous approach involved misstating the property if $B \subseteq A$ then $B…
fesodes
  • 61
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If $(a_n)\to\infty$, then $(1/a_n)\to0$

Prove: If $(a_n)\to\infty$, then $(1/a_n)\to0$ As $a_n\to\infty$, for every $C>0$ there exists an $N$ in the natural numbers such that we have $a_n>C$ whenever $n>N$ $\implies|a_n|>C$ whenever $n>N$ $\implies|1/a_n| < 1/C$ whenever $n>N$ If…
James Doherty
  • 127
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2 answers

Primary ideals of $\mathbb{Z}$

Let $R=\mathbb{Z}$ I must prove that In $\mathbb{Z}$ the primary ideals s are precisely the ideals $(p^k)$ where $p$ is a prime number and $n\ge 1$. $$$$ The definition of primary ideal. Definition. An ideal $I$ of the ring $R$ is called primary…
Jack J.
  • 920
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1 answer

Proving the inverse of a function $f$ is a function iff the function $f$ is a bijection.

Can someone give me their opinion on how I wrote this proof also all my definitions are at the end. Given a function $f:A\to B$, its inverse relation $f^{-1}$ is a function from $B$ to $A$ if and only if $f$ is a bijection from $A$ onto…
George
  • 117
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0 answers

Questions about nilpotent rings

My question concerns some steps of a proof. Let's start with some definitions. Definition 1. Let $R$ a ring, an element $x\in R$ is said to be nilpotent if exists $n\in\mathbb{N}\setminus\{0\}$ such that $x^n=0$. Definition 2. An ideal $I$ of a…
Jack J.
  • 920
2
votes
2 answers

Prove that if $p, p+2$ and $p+4$ are primes, then $p=3$

Since $p+2$ and $p + 4$ are prime, $3$ doesn't divide either of them. But then $3 \mid p+3$, which implies $3 \mid p$. Since $p$ is prime, then $p = 3$. Is it this simple?
Matheus Andrade
  • 6,773
2
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1 answer

Show that $\log_b{(y\cdot z)} = \log_b(y)+log_b(z)$

For $a,b > 1$ and $y,z > 0$, show by using only the power rules and the definition $x = \log_b(y) \Leftrightarrow_{df} y = b^x$ that $\log_b{(y\cdot z)} = \log_b(y)+log_b(z)$. I don't get how to solve this without using $b^{log_b{x}} = x$, which…
WhatAMesh
  • 871
2
votes
1 answer

Inductive Proof on String Inequality

I'm pretty new to (and therefore bad at) inductive proofs; I'm trying to do a problem from The Formal Semantics of Programming Languages, which reads: Using mathematical induction, show that there is no string $u$ which satisfies $au = ub$ for…
Geoff Huston
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