+2499 
Trying to find out if these are equal, I don't know how to open the brackets.
+118687 Hi Solveit,
Perhaps someone can tell me what the U symbol is, I have not seen it before. 
+118687 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\\~\\
