If $M=A \cup B$ is $A \in M$, $A \subset M$, or both? The distinction between these two symbols has always eluded me. Can someone please explain?
 A: It is always the case that $A \subset A \cup B$ because if $x \in A$ then $x \in A \cup B$.
It may be the case that $A \in A \cup B$, for example, $A = \{1\}$ and $B= \{ \{ 1 \} \}$. In this example, $A \in B$, hence $A \in A \cup B$.
Generally we try to avoid such things.
A: First, some basic symbols:


*

*$A\in B$ means $A$ is an element of $B$ - that is, $A$ is part of the set $B$. 

*$A\cup B$ means the union of the sets $A$ and $B$ - that is, all of the objects that are part of the set $A$, the set $B$, or both.

*$A\subset B$ means that $A$ is a subset of $B$ - that is, all of the elements in $A$ are also in $B$. (It should be noted that every set is a subset of itself; i.e., $A\subset A$.)


Some more basic symbols are given here.
Now, for your specific question. If $M = A\cup B$, that means $M$ contains all of the elements of $A$ and $B$. Therefore, $A$ is indeed a subset of $M$ as all of its elements are by definition in $M$, so $A\subset M$. Now, lists aren't really elements. So, if $A = \{1,2,3\}$, then $A\notin B$ ($\notin$ means $A$ is not an element of $B$). However, if $A$ only has a single element, for example if $A=\{1\}$, then $A\in B$, but this notation should be avoided.
Sometimes it can help to draw a venn diagram:

represents $A\cup B$ and from here we can clearly see that $A$ is indeed a subset of $M$.
Hope this helps!
A: $A \subset M$ means $A$ is a subset of $M$.
Examples:  $\{Peter, Paul, Mary, Luke, John\}\subset ${people mentioned in  the bible}.  Because every element of the $\{Peter, Paul, Mary, Luke, John\}$ are each members of {people mentioned in  the bible}. 
$A \in M$ means $A$ is an element of $M$.
Example $Peter \in ${people mentioned in the bible}
$\{Peter, Paul, Mary, Luke, John\} \not \in ${people mentioned in  the bible} because ..... $\{Peter, Paul, Mary, Luke, John\}$ is not a person mentioned in the bible.  THere is no person whose name is "$\{Peter, Paul, Mary, Luke, John\}$"
Example: $\{\text{even numbers}\}\subset \mathbb Z$  because the set of all even numbers is a smaller set that is completely contained in the integers.
$2 \not \subset \mathbb Z$ because $2$ is not a set at all; it's an integer.  (Oh, be quiet, you set theorists!  You can tell me all about how from a set theoretical construct of axioms everything is a set and what an idiot I am somewhere else.)
$\{\text{even numbers}\}\not \in \mathbb Z$ because $\{\text{even numbers}\}$ is not an integer.  It's a set of integers.  
Basically a set is a list and a subset is a smaller list.  Members are the things in the list.  A shopping list are a list of the things you will buy.  The list, itself, is not one of the things you will buy.  If you try to eat your shopping list... it won't work.
So $M = A \cup B$.  Is $A$ an element in the set $M$? or is $A$ a smaller set composed of some of the elements of the set $M$.
