Ad4mWayn3

I have some familiarity with C++, and concepts like compiling and linking static and dynamic libraries, which is what I understand as collections of code that simplify doing certain things.

But then I get confused in certain cases, for example, why is OpenGL considered an API? Why is it necessary to use other libraries like GLAD, freeGLUT or GLFW to interface with OpenGL?

And then other languages have this thing called package managers, like pip, node, cargo, and vcpkg for c/c++, where you install these packages that get used like libraries? What's the difference?

Finally the ones I understand the least of are frameworks. I keep hearing the concept of frameworks like Angular for js and a lot of stuff that's too alien for me because I'm still unfamiliar with web development.

So for example, I'm using the raylib library for a small game project I have. I link the .lib or .dll file to my executable file so I know I'm unambiguously using a library. How come there's also Cocos2dx which is a framework? What's the distinction?

Given two real, nonzero algebraic numbers a and b, with a > 0 (so that it excludes complex numbers), is there any named subset of the reals S such that (a^b) belongs to S forall a,b? I know it's not all the reals since there should be countably many a^b's, since a,b are also countable.