# Tagged Questions

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

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### How to proof that the set of all $X$ such that $X.A{\ge} c$ to some real number c is convex?

How can i proof the following statement: " Let $\mathrm A\in \mathbb R^{n}$ and $\mathrm c\in \mathbb R$, the set $\mathbb S$ of all elements belonging to $\mathbb R^{n}$ and satisfying the ...
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### Prove $\forall n \geq 10, 2^n > n^3$

Prove $\forall n \geq 10, 2^n > n^3$ base case: $n = 10$ $2^{10} = 1024$ $10^3 = 1000$ $1024 > 1024$. So $P(k)$ holds for $k = n$. We seek to show $P(k+1)$ holds: We know $2^k > k^3$. ...
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### Prove $f: A \rightarrow B$ is strictly injective, $\implies$ $f^{-1}$ is a function and dom $f^{-1} \subset B$

The question I have about this proof is that, do I need to choose a specific function $f:A\rightarrow B$ that is not injective but surjective? Will I lose generality if I do? For instance, I was ...
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### If $f$ is injective, then $f(X\backslash A) = f(X) \backslash f(A)$

Given $f:X \to Y$ injective, $A \subseteq X$, then $f(X\backslash A) = f(X) \backslash f(A)$ I have spent a long time looking at this problem but I have not found a good way to approach this. Here ...
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### For $f: \mathbb{N} \rightarrow \mathbb{N}$ given by $f(x)=ax+b$, for what $a,b \in \mathbb{R}$ is $f$ a bijection?

I asked a similar question here. This question has different parameters however as you can see. For $f: \mathbb{N} \rightarrow \mathbb{N}$ given by $f(x)=ax+b$, for what $a,b \in \mathbb{R}$ is $f$ a ...
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### Nullity and an Isomorphism

I'm working on some introductory proofs in linear algebra, and I think that I could use some help on this particular problem. I want to prove that a linear surjective map $T: R \rightarrow W$ is an ...
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### Prove that $2^n$ does not divide $n!$

I want to prove that $2^n$ does not divide $n!$. I was trying by induction and I'm confused about if what I'm doing is right. First I test it with $n=1$. In fact: $$2^1 \nmid 1!$$ So if i take the ...
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### Any additional constraint will increase the value of the maximin

In the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $325$, there is the comment "any additional constraint will increase the value of the maximin",...
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### Prove that $\int_{a}^{b} f > 0$

I am asked to prove that $\int_{a}^{b} f > 0$. we are given that $f$ is continuous on $[a,b]$ $(\forall x \in [a,b]) \; f(x) \geq 0\;$ and $(\exists x_{0}) \in [a,b] \;s.t.\; f(x_{0}) >0$ my ...
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### Proof Related to the Span in linear algebra

I'm working through a proof in my linear algebra textbook, and I think I am a little stuck. I am trying to prove that if $S$ is a non-empty set of vectors in a vector space $V$, the the set $W_s$ of ...