- Like you said, $a \propto b$ means: a is proportional to b and so:
$a \propto b\Rightarrow a=k\cdot b$ for some $k \in \mathbb{R}$
- In contrast: $a \sim b$ means: a is distributed according to b and so:
$a \sim b \Rightarrow \lim_{x\rightarrow \infty} \dfrac{a(x)}{b(x)} = 1 $
So, $\sim$ indicates identical asymptotic behavior for arbitrary functions $a,b$. Maybe functions over time for your usecase?
Note that although $a \propto b$ is a more general statement than $a \sim b$ since it holds for all values of $a$ and $b$, rather than only for the limit, it does not imply $a \sim b$.
Well, a circle can be described by a function, just not in the sense that you may be familiar with. If you are looking at a function that describes a set of points in Cartesian space by mapping each $x$-coordinate to a $y$-coordinate, then a circle cannot be described by a function because it fails what is known in High School as the vertical line test.
A function, by definition, has a unique output for every input. However, for almost all points on a circle, there is another point with the same $x$-coordinate. So, you would need your function to give two different $y$-coordinates for certain inputs, which is not allowed.
However, there is no rule that the input of a function has to be an $x$-coordinate or that the output has to be a $y$-coordinate, so we can define other functions that describle a circle. In more formal terms, the domain and codomain of a function do not have to be $\Bbb{R}$. For example, we can have a function that outputs an ordered pair (that is, codomain of $\Bbb{R}\times\Bbb{R}$). Then, $$f(t)=(\sin t,\cos t)$$ outputs the unit circle when $0\le t<2\pi$. We could also describe the points in space in a different way, using polar coordinates. Here we use the counter-clockwise angle from the positive $x$-axis, $\theta$, and the distance from the origin, $r$, to identify a point. Using this system, we can easily describe the unit circle as $(\theta,f(\theta))$, where $f(\theta)=1$ and $0\le\theta<2\pi$.
Best Answer
They are the same in a sense. A function is technically an "equation", but an equation is not necessarily a function. It is incontrovertibly true; however, to say that any two values separated by an equals sign is considered an equation. $5=2+3$ is an equation, but not a function. $y=2x+3$ or equivalently, $f(x)=2x+3$ is a function, but also an equation because it involves an equals sign. Functions are equations that have both constants and variables which arrange in some way to map the function's value about those variables' dimensions.