Question: ![enter image description here](https://i.stack.imgur.com/lCZBF.png)

$$\bigcup\limits\_{n\in\mathbb N} A\_n$$

The book is asking me to prove that $f(\bigcup\limits\_{n\in\mathbb N} A\_n) = \bigcup\limits\_{n\in\mathbb N} A\_n$.

I'm able to prove that f(the notation above) = the notation above U A(n+1) (A(n+1) looks like An above)
does that mean that f(the notation above) = the notation above? since n+1 is in N

Does this notation mean the union of the sets $A\_1 \cup A\_2 \cup ... \cup A\_{6565656}$?

Answer: 
There is a [relation](http://mathworld.wolfram.com/VietasFormulas.html) between the coefficients of a polynomial equation of degree $n$ and its roots.

> 
>  It follows easily that $ x, y, z $ are the roots of the equation 
>  $$ p^3 - 6p^2 + 11p -6 = 0 \Leftrightarrow (p-1)(p-2)(p-3) = 0, $$ 
>  which are $ p = 1, 2, 3 $. The equations of the system are symmetrical in $ x, y, z $. Then, the six solutions of the system are the permutations of $ 1, 2, 3 $: *i.e.*, $$ (x, y, z) = (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1). $$
> 
> 
>
With the given info, $x,y$ and $z$ are roots to the polynomial
$$
s^3-6s^2+11s-6=s^3-3s^2-3s^2+9s+2s-6\\
=(s-3)(s^2-3s+2)=(s-3)(s-1)(s-2).
$$