Which one is the correct notation?

Question

I know this notation is correct:

$a_1,a_2, a_3,\cdots,a_n=\left\{a_k\right\}_{k=1}^n$

Now, we have a function $f(n)$.

I want to write this sequence in correct notation:

$\left\{ f(1),f(2),f(3),\cdots ,f(n)\right\}$

I have two notations. Which one is correct?

$\bigcup_{k=1}^nf(k)=\left\{ f(1),f(2),f(3),\cdots , f(n)\right\}$

$\left\{f(k)\right\}_{k=1}^n=\left\{ f(1),f(2),f(3),\cdots, f(n)\right\}$

Thank you.

Answer 1

In

$$\left\{a_k\right\}_{k=1}^n,$$ $a_k$ can be anything. For instance $f(k)$.

But $$\bigcup$$ is a set operator, so that its arguments must be sets, which the $f(k)$ are probably not. Even if you used $\{f(k)\}$, you would still obtain a set instead of a sequence (a set is unordered).