Minimum function

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Needs to be linked to order theory stuff. Also:

• Characteristic property of the minimum: $$\forall x,u,v\in S\big[x\eq\text{min}(u,v)\iff\big(x\preceq u \wedge x\preceq v \wedge (x\eq u \vee x\eq v)\big)\big]$$
  • Proved for $\implies$ direction.
  • I was working on the $\impliedby$ direction I was about to (attempt) to prove this lemma (and then use it)
    • Unproven lemma: $$\big(x\preceq u\wedge x\preceq v)\implies x\preceq \text{min}(u,v)$$ - via contrapositive
      • Note to self: be sure the logical and has a negation of logical and link ect for all operators! That's a long time coming!
• Some useful lemma: $$\forall x,u,v\in S\big[x\eq\text{min}(u,v)\implies\big(x\preceq u\wedge x\preceq v\big)\big]$$ which I (believe) I used in defining the minimum of random variables.

This page is short because I wrote it just prior to bed Alec (talk) 07:53, 28 July 2018 (UTC)

Addendum: I will add the following definition:

Let $(S,\preceq)$ be a "totally ordered poset" ("oset"?)

TODO: Look into this

- it must be totally ordered so $\forall x,y\in S$ exactly one of $x\prec y$, $x\succ y$ or $x\eq y$ holds ("trichotomy law" or something)

Then:

• $$\text{min}:S^2\rightarrow S$$ is a function defined by $$\text{min}(u,v)\mapsto\left\{\begin{array}{lr}\pi_1(u,v) & \text{if }u\prec v\\ \pi_2(u,v) & \text{if }u\succ v \\ \alpha & \text{otherwise}\end{array}\right.$$ where $\pi_1$ and $\pi_2$ are the characteristic projections of a product, and
  • $\alpha:\eq \pi_{\alpha'}(u,v)$ for $\alpha'\eq 1$ or $\alpha'\eq 2$ (it doesn't matter as in the case where $\alpha$ is used we have {$u\eq v$ by the "trichotomy law" mentioned above.
  • We include $\alpha$ to make the cases in analysis more explicit (as this is a Category:First-year friendly page, meaning the readers' hand is held as he reads the steps involved) but also because there is a practical case for a kind of (computer) arithmetic where it is possible that we have $\alpha\neq u$ and $\alpha\neq v$ - It might be signed zero, I remember doing it, not why I was doing it unfortunately.