Let $X$ and $Y$ be arbitrary sets with arbitrary elements $x\in X$ and $y\in Y$. Let $Y'\subset Y$. Further, let $f:X\times Y\to\mathbb{R}$ be some function and let $f(x,Y')=\{f(x,y)\mid y\in Y'\}\subset\mathbb{R}$. Thus, $f(x,Y')$ is a set of Real numbers.

Then, if $f(x,Y')$ is the singleton containing just $z\in\mathbb{R}$, we can simply write $f(x,Y')=\{z\}$. If $f(x,Y')$ is a singleton whose unique element is equal or greater than $z\in\mathbb{R}$, I have thought of writing $f(x,Y')\geqslant\{z\}$.

Unfortunately, I think my notation is wrong, because (unlike the relation $=$) the relation $\geqslant$ is usually only defined for numbers rather than sets of numbers (even if those sets are singletons!). I have thought of using the expressions $\min_{y\in Y'}\{f(x,y)\}\geqslant z$ or $\min\{f(x,y)\mid y\in Y'\}\geqslant z$, but that seems too cumbersome for such a simple idea.

Although, perhaps, the best is simply to write $f(x,y)\geqslant z$ for all $y\in Y'$, isn't it?

What would you do?

Any help will be much appreciated.

  • 1
    $\begingroup$ One thing I’d do is be sure what the codomain is: you write $f\to\Bbb R^{\color{red}{X}}$, but elsewhere in the post treat $f\to\Bbb R$. I might write $\inf f(x,Y’)\ge\sup f(x,Y’’)$. However, it’s ok to use plain words! If you read a paper or a textbook, you’ll see that a lot of content which could be written symbolically - isn’t. $\endgroup$
    – FShrike
    Commented Aug 19, 2022 at 8:52
  • $\begingroup$ If the codomain is $\Bbb R^{X}$ then: $\{f(x,y):y\in Y’\}$ is not a subset of $\Bbb R$, yet you treat it to be $\endgroup$
    – FShrike
    Commented Aug 19, 2022 at 9:00
  • $\begingroup$ Sorry, of course. Error now corrected. $\endgroup$
    – EoDmnFOr3q
    Commented Aug 19, 2022 at 9:02
  • $\begingroup$ There is at least 1 more error : If $ Y` $ gives Singleton $ \{z\} $ & $ Y $ also gives Singleton $ \{w\} $ , then these two Singletons must be same $ z=w $. Instead, you could use some other $ Y`` $ , where $ Y` $ & $ Y`` $ will give Distinct Singletons. $\endgroup$
    – Prem
    Commented Aug 20, 2022 at 9:13


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