# How to write that the number in a singleton is larger than some other number?

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.

• 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. Commented Aug 19, 2022 at 8:52
• 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 Commented Aug 19, 2022 at 9:00
• Sorry, of course. Error now corrected. Commented Aug 19, 2022 at 9:02
• 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.
– Prem
Commented Aug 20, 2022 at 9:13