Consider these three sets $(0\notin\mathbb{N}$ for the sake of this example):
$M_1:= \{n\in\mathbb{N}:n $ is even$\}$
$M_2:= \{n\in\mathbb{N}:n $ is odd$\}$
$M_3:= \{1,2,3,4,5\}$
I need to find $(M_1\cap M_2)\cup(M_3\cap M_2)$ which I did, but I have some problems with the notation regarding the empty set.
Version 1:
$(M_1\cap M_2)\cup(M_3\cap M_2)=\emptyset\ \cup \{1,3,5\}=\{\emptyset,1,3,5\}$
Version 2:
$(M_1\cap M_2)\cup(M_3\cap M_2)=\emptyset\ \cup \{1,3,5\}=\{1,3,5\}$
Version 3:
$(M_1\cap M_2)\cup(M_3\cap M_2)=\emptyset\ \cup \{\{1\},\{3\},\{5\}\}=\{\emptyset,\{1\},\{3\},\{5\}\}$
Which is the correct notation?
Only version 2 is correct. For any set $A$ we have $A\cup \emptyset=A$.
Also note: $M_3\cap M_2$ is a set of numbers (because $M_2$ and $M_3$ are). Hence the elements of $M_3\cap M_2$ are numbers, not (singleton) sets of numbers. This makes version 3 wrong.