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### Notation for choosing the minimum of a set after applying the norm

I have a function $x: \mathbb{R^+} \mapsto \mathbb{R}^n$ defined as $x(k) = \text{min}\{p \in S: \lVert p \rVert > k\}$ (where $S \subseteq \mathbb{R}^n$). I want to choose the point with the ...
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### Discrete Math Proof of Logics

In a question, given that $P(n)$ is a statement for integers $n \geq 1$ such that for all $n : P(n) \implies P(n+4)$ . How can I determine if $$P(1) \wedge \neg P(4n + 1)$$ is true or false.
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### The group of all bijective functions on S with composition as its binary operation is finitely-generated iff S is a finite set

Here is the problem, I thought it the I've been thinking for a long time, but I still don't have any ideas. Problem: Let A(S) be the group of all bijective functions on S with composition as its ...
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### Lebesgue-Stieltjes Measure on $\Bbb R^2$

If $(\Bbb R,B(\Bbb R),\mu)$ is measure space such that $\mu$ is finite on all bounded set in $B(\Bbb R)$.Then it correspondence to family $\mathcal A$ of non-decreasing and right continuous real ...
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