Meaning of math symbol ~

Question

Segment of Example from Textbook:

$t=\dots$

More usefully, we have:

$t\sim n\log n$

I recall $\sim$ means "similarity" in geometry, same shape but not same size. How is it interpreted here?

Answer 1

In this context, it means that $$\lim_{n \to \infty}\frac{t}{n\log n}=1$$ That is, the quotient of both sides tends to $1$.

Answer 2

The answer by Daniel Littlewood is absolutely correct in this context. To extend this to the general definition (not tied to the example at hand):

$$f(n) \sim g(n) \iff \lim_{n\to\infty} \left(\frac{f}{g}\right)(n) = 1$$

Answer 3

The symbol $\sim$ does not have a set meaning across all subjects, but it is almost always used to denote an equivalence relation: a relation that is reflexive, symmetric, and transitive.

Daniel Littlewood and anorton have already discussed what $\sim$ means in this instance, and we can verify that it is an equivalent relation between functions on $\Bbb R$.

Clearly for any function $f(n)$, we have that $f\sim f$ since $(f/f)(n)=1$ for all $n$ (with caveats about the zeros of $f$) and $\lim_{n\rightarrow\infty}1=1$.

Also if $f\sim g$ we then have that $g\sim f$. This follows from the fact that if $\lim_{n\rightarrow\infty}h(n)=1$ then $\lim_{n\rightarrow\infty}(1/h)(n)=1$. Applying this fact with $h=f/g$ we tell us that $\sim$ is symmetric.

Finally $\sim$ is transitive. This follows from the fact that if $\lim_{n\rightarrow\infty}h_1(n)=1$ and $\lim_{n\rightarrow\infty}h_2(n)=1$ then $\lim_{n\rightarrow\infty}(h_1h_2)(n)=1$. If we have that $f\sim g$ and $g\sim h$, apply the previous fact with $h_1=f/g$ and $h_2=g/h$, and you will get that $f\sim h$.