Is this the correct way of using ~?
Well, notation usually depends on the guy who is giving the lecture, or the guy who wrote the book. There are some symbols that are used more frequently for the same intention. So you could say, that they are some kind of a standard notation.
A notation that most people will feel comfortable using. A notation that if used most people automatically know what you mean.
E.g. most people use $+$ for addition. That sounds stupid, but there are not only "numbers" in math and there are structures where addition is differently defined as the "$+$" we are used to. So, if you read $$a \oplus b = 42$$
in a book, you will automatically question yourself what special operation does the author denote by $\oplus$. Since he didn't use $+$ it might be something special.
Your question will lead to very strongly subjective answers. Since most people tend to acquire their own notation over the years. For example, even though I do numerical mathematics, and I am definitively not a pure math guy, I will reserve some symbols for pure math stuff. I will rarely use them, but I was lecture assistant for a linear algebra course once. So, that got stuck with me.
I, personally, would not use $\sim$ for your purpose, but rather $\approx$, like $$π\approx 4.$$ The reason is simply, that I reserve $\sim$ for the purpose of relations and proportionality. Both are mathematical concepts. If you have never used these two concepts, or if you don't need them often, or if you choose to overload $\sim$, because you don't care too much, go ahead and use $\sim$ for your purpose.
Nevertheless, $\sim$ is often used for that purpose by non-mathematicians. Probably because it is faster to write $\sim$ than $\approx$.
Here is an example for proportionality:
$$\text{number of steaks}\sim\text{number of cows},$$
meaning the number of steaks is proportional (proportionally growing) with the number of cows.
For relations an example is: If "person $x$ is sitting on the same table as person $y$", then we write $x\sim y$ or say "$x$ is related (in the way defined before) to $y$".
About $\backsimeq$ and $\cong$: I personally use both for the term isomorphism.
I don't want to go into detail, but I will try to give you a rough idea.
One usually knows mappings only as functions like:
$$f:ℝ→ℝ\qquad x↦f(x)=x^2$$
So this function maps objects of $ℝ$ to objects of $ℝ$. It maps numbers to numbers. In this case $ℝ$ is just some pile of things - of numbers. That is nothing special.
Now mathematicians like structure, they like it a lot - it's great, you will love it. (sorry, couldn't resist ^^)
Therefore, one of the first things you look at are piles of things with a structure on it. For example there might be connection between these things, e.g. "addition". Now you can connect two things of that pile, and (maybe) will result in something that is in that pile, like $3+6=9$.
Or there might be one special thing, called neutral element, that doesn't change other things if you use the connection, like $42 + 0 = 42$.
To give them correct mathematical names: A "pile" is called set, and with some specific structure you will get for example a group.
E.g. the set of positive and negative integers $ℤ=\{0,1,-1,2,-2,…\}$ together with the connection $"+"$ is such a group.
Now imagine you have two groups $G$ and $G'$, two different piles with structure. The next natural step is, to connect both groups. You do that with a mapping:
$$φ:G→G'$$
Because mathematicians like to name things, they call these mappings "homeomorphisms", if they preserve the structure:
$$φ(x+_Gy) = φ(x)+_{G'}φ(y).$$
That means it does not matter if you first connect two elements $x,y$ of $G$ and then map them to $G'$ or if you map them to $G'$ first, resulting in $φ(x)$ and $φ(y)$ and then connect them in $G'$ with the addition $+_{G'}$ of $G'$. These mappings help a lot, e.g. one conclusion is that
$$φ(0_G) = 0_{G'}$$
So they map the neutral element of $G$ to the neutral element of $G'$.
If that mapping is even more special it is called an isomorphism. And to denote that there exists some isomorphism between two group you write $G\cong G'$.
I hope my answer helped you.
~out in the world, but not in mathematics. – Arthur Aug 16 '17 at 08:16