# Tagged Questions

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

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### A set A is infinite if and only if there is a bijection from the set A to a proper subset of A. [duplicate]

I'm just starting my journey into proof writing and I don't really know how to do this. More specifically I think I want a proof of the fact that every infinite set A is Dedekind-infinite (i.e. that ...
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### Implication vs Equivalence in proofs

I understand the definition of both the implication and equivalence signs. When I get asked to prove something, I will probably have to use both implication and equivalence logic. My question is if it'...
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### Graph theory: proving that a graph with specific property is bipartite

I have been given the following problem on an exercise sheet: Let $G$ be a graph with $n$ vertices with the property that for each $k ≤ n$, every set of $k$ vertices contains a subset of size at ...
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### Proving the area of a triangle within a triangle

Consider a triangle with vertices ABC, we pick a point C' on the line segment AB in such a way that |BC'|=2|AC'|. Similarly, we pick a point B' on the line segment AC and a point A' on the line ...
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### Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular, without using $\lambda_1\cdot\ldots\cdot\lambda_n = det(A)$. [duplicate]

I would like to know if there is any proof without using the fact that: $$\lambda_1\cdot\lambda_2\cdot\ldots\cdot\lambda_{n-1}\cdot\lambda_n = det(A)$$ I managed to prove that if $\lambda = 0$ then, ...
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### Prove that minimum spanning tree is a tree

From the the Wikipedia page Minimum spanning tree: A minimum spanning tree is a spanning tree of a connected, undirected graph. It connects all the vertices together with the minimal total ...
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### Proving a group homomorphism: $\phi: G\rightarrow \text{Aut}(G), g\mapsto g^*$

Given $\phi: G\rightarrow Aut(G), g\mapsto g^*$ and $g^*: G\rightarrow G, x\mapsto gxg^{-1}$ where $g,g^{-1},x\in G$ I need to prove that $\phi$ is a homomorphism ($\phi(gh)=\phi(g)\phi(h)$). So, ...
### Proof that the exponential function is continuous on $\mathbb{R}$ without use of derivatives
I am still trying to understand how to prove statements. I want to prove that for $a>0$, $f(x) = a^x$ is continuous on $\mathbb{R}$. The text gives an hint, namely, that it suffices proving ...