The \sim command provides the basic similarity symbol (like ~), which is a predefined relational symbol in standard math mode. The symbols can be used for similar or asymptotically equal (i.e., approximating to), congruent to, approximately equal, and more.
The \approx command provides a symbol that’s meant to show something is “approximately equal.” The \equiv command shows something is “identical to.”
Proportionality belongs to this same family of symbols; you’ll find it under the name \propto.
In addition to the commands listed above, there are also the commands \lesssim (less than or similar to) and \gtrsim (greater than or similar to).
All of these commands work in math mode with either a $ sign or by using an equation environment such as align. All of the following commands (\sim, \simeq, \cong, \approx) are part of the standard TeX package and available out-of-the-box. However, some extended versions—such as \lesssim and \gtrsim—are not and require importing the amssymb package.
If you’re unsure what symbol you want, try Detexify, a free online tool where you can draw a symbol on your screen, and it will return the corresponding LaTeX command. Symbols may have multiple meanings depending upon their context.
Similarity and related symbols
LaTeX provides several symbols related to the basic symbol for similarity (~). In many cases, they mean essentially the same thing, though one might be more appropriate than another in specific circumstances. Let’s walk through each one:
The basic symbol is provided by the command \sim. It is a relational symbol and is predefined in TeX and thus available without any special package. Its symbol is simply ∼. As the name suggests, the symbol expresses that two things are “similar,” meaning that they have the same shape or structure but aren’t necessarily exactly identical.
There are other symbols that express similar ideas. The symbol for “asymptotically equal” is provided by the command \simeq. The resulting symbol is ≃. This means that two expressions get closer and closer to being equal as variables approach infinity or something else.
Two symbols indicate that two expressions are “congruent”: \cong, producing the symbol ≅, and \approx, producing the symbol ≈. The difference between the two is rather subtle: The former indicates that the expressions are essentially identical, while the latter implies that they are merely close enough.
There are also a few symbols that combine a relation and similarity. For example, the command \lesssim produces the symbol ⪯, which means that a value is less than or similar to another. Similarly, the command \gtrsim produces the symbol ⪰, which is greater than or similar to.
Proportionality belongs to this same family of symbols, though it doesn’t seem to express a relationship in quite the same way. The command \propto provides the symbol ∝, which means “is proportional to.” You’ll find this command listed alongside \sim and \cong as one of the predefined TeX relational symbols.
Finding the right command
None of the above commands should cause problems for you if you just type them into your LaTeX document. If you’re ever unsure what the correct command is, try drawing the symbol on your screen. If you do that, then the site Detexify will tell you what the symbol is called.