What is the difference between $\{1, 2\}\cup\{3\}$ and $\{1, 2, 3\}$? I have a simple question related to set theory. 
Is there any difference between the following two sets?
What is the difference between $\{1, 2\}\cup\{3\}$ and $\{1, 2, 3\}$?
 A: There is no difference; they are the same set.
A: Others have already pointed out that the sets are the same.
Remember that two sets are the same when they have the same elements. That is, two sets $X$ and $Y$ are the same exactly when $a$ is an elements of $X$ if and only if $a$ is an element of $Y$.
So consider the two sets $X = \{1,2\}\cup \{3\}$ and $Y= \{1,2,3\}$. The elements of $X$ are exactly $1, 2,$ and $3$. The elements of $Y$ are exactly the same. So the two sets are equal.
A: The way that we show two sets are equal is by showing that every thing in one set is in the other set, and vice versa. 
i.e. $$A=B \text{ if and only if } A\subseteq B \text{ and }B\subseteq A$$
Since union is defined to be the set of elements in both sets in the union, it is obvious in your situation that the two sets in question are equal.
A: The union operator $\cup$ joins the sets $\{1, 2\}$ and $\{3\}$ to form the set $\{1, 2, 3\}$. So
$$
\{1, 2\} \cup \{3\} \equiv \{1, 2, 3\}.
$$
A: Since the sets are equal, the difference (in either direction) is equal to $\varnothing$.
A: The union of $\{1,2\}$ and $\{3\}$ is equal to $\{1,2,3\}$.  The intersect of $\{1,2\}$ and $\{3\}$ would be equal to $0$.
