Are these equal?

Hi Solveit,

Perhaps someone can tell me what the U symbol is,  I have not seen it before.   

Solveit tells me that U just means union.

I have never seen it used like this before so anything I write is pure guess work.

It seems to me that 

   \(\displaystyle \cup _{n=1}^{5}\;A_n = \{A_1, A_2, A_3,A_4, A_5\}\)

\(\displaystyle \cup _{k=1}^{n}\;A_n = \{A_1, A_2, A_3,A_4, A_5, ....., A_n\}\\~\\ (\displaystyle \cup _{k=1}^{n}\;A_n) = \{A_1, A_2, A_3,A_4, A_5, ....., A_n\}\\~\\ \displaystyle \cup _{k=1}^{5}(\displaystyle \cup _{k=1}^{n}\;A_n) = \{A_1\}\cup \{ A_1, A_2\} \cup \{ A_1, A_2, A_3\} \cup \{ A_1, A_2, A_3, A_4\}\cup \{ A_1, A_2, A_3, A_4,A_5\}\\~\\ \displaystyle \cup _{k=1}^{5}(\displaystyle \cup _{k=1}^{n}\;A_n) =\{ A_1, A_2, A_3, A_4,A_5\}\\~\\ \displaystyle \cup _{k=1}^{5}(\displaystyle \cup _{k=1}^{n}\;A_n) =\displaystyle \cup _{k=1}^{5}\;A_n\\~\\\)

LaTex:

\displaystyle \cup _{k=1}^{n}\;A_n = \{A_1, A_2, A_3,A_4, A_5, ....., A_n\}\\~\\
(\displaystyle \cup _{k=1}^{n}\;A_n) = \{A_1, A_2, A_3,A_4, A_5, ....., A_n\}\\~\\

\displaystyle \cup _{k=1}^{5}(\displaystyle \cup _{k=1}^{n}\;A_n) = \{A_1\}\cup \{ A_1, A_2\}  \cup \{ A_1, A_2, A_3\} \cup \{ A_1, A_2, A_3, A_4\}\cup \{ A_1, A_2, A_3, A_4,A_5\}\\~\\
\displaystyle \cup _{k=1}^{5}(\displaystyle \cup _{k=1}^{n}\;A_n) =\{ A_1, A_2, A_3, A_4,A_5\}\\~\\
\displaystyle \cup _{k=1}^{5}(\displaystyle \cup _{k=1}^{n}\;A_n) =\displaystyle \cup _{k=1}^{5}\;A_n\\~\\