# Questions tagged [alternative-proof]

If you already have a proof for some result but want to ask for a different proof (using different methods).

1,910 questions
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### Proof for upper bound of function using an approximation through Squeeze Theorem.

I am trying to turn $n_0log_2(n_0)$ into an approximation of $nlog_2(n)$ using the following statement: $$n \geq n_0+1$$ For a bit of context, $n$ is the number of nodes on a minimum spanning tree [...
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### Let $A$ be a square matrix. Prove that $A$ ~ $I_n$ if and only if $A\vec{x} = \vec{0}$ has only the trivial solution.

I'm studying Linear Algebra for the second time, using Hoffmann & Kunze. Currently I'm trying to prove the following theorem: Theorem 7. If $A$ is an $n \times n$ matrix, then $A$ is row-...
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### On exercise 14H from Willard's 'General Topology' book

$\newcommand{\R}{\mathop{\mathbf R}} \newcommand{\FN}{\mathop{\mathfrak N}}$ The exercise is in p. 99 of the book. It says the following: Let $X$ be a topological space, and let $B(X,\R)$ denote ...
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### Prove $(\lnot p \to p) \to p$ in 19 steps

While revisiting "Mathematical Logic", I noticed a quiz that I had never worked out. It requires to prove $$\{\} \vdash (\lnot p \to p) \to p$$ in 19 steps. To be clear, each "step" should be in ...
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### Elements and cyclic subgroups of order $15$ in $\Bbb Z_{30}\times \Bbb Z_{20}.$

This is Exercise 8.22 of Gallian's, "Contemporary Abstract Algebra". Please use only methods from this book prior to the exercise. This is an alternative-proof question. Find the number of ...
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### Proof of exponential law using limit definition of exponential function?

For fun, I tried to prove the well-known exponential property $e^{a+b} = e^a e^b$ using the limit definition of the exponential function, below. Definition. The exponential function is defined as ...
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### Is there a proof of quadratic reciprocity using $p$-adic numbers?
I know that the quadratic reciprocity can be regarded as a special case of Artin reciprocity (class field theory), and we can get it by considering the cyclotomic extension of $\mathbb{Q}_{p}$. ...