As it is virtually impossible to list all the symbols ever used in mathematics only those symbols which occur often in mathematics or mathematics education are included. Union and Intersection symbols in LaTeX can be produced via the cup and cap definitions while in math mode. Amsmath amssymb or mathtools.
You should use bigcuplimits_i1infty F_i but in inline math mode its better to set the limits on the side just dont use limits which is implicit in display math mode.
1 Class 0 Ord symbols. This is not a comprehensive list. If you use xelatex and unicode-math you can simply use the symbol directly or its alias cupdot. LaTeX Math Symbols The following tables are extracted from The Not So Short Introduction to LaTeX2e aka.