Q:

Set Theory question about equalities of sets

Consider the following sets: 
$A=\{3,4\},$     $B=\{4,3\}\bigcup \emptyset, C=\{4,3\}\bigcup\{\emptyset\}$
Which pairs of sets are equal? 
My attempt: We obviously have $C=\{4,3,\emptyset\}$ We also have(?)  $B=\{4,3\}$ as by the Null set Axiom, the empty set is the unique set having no elements. 
Now by the extensionality axiom, $A$ and $B$ are equal since they have the same elements.  However, they are not equal to $C$ as they don't contain the empty set as an element. 
Is that true?  I'm still unsure about the bit where I look at the empty set as an element of $C$ since technically, no element of $C$ is not an element of $A$ or vice versa, hence we can use ZF1 again to conclude that they are equal?? 
Thanks in advance

A:

You are correct. 
Except that to properly conclude that $A\ne C$ (and likewise $B\ne C$) you need to show that $\emptyset\notin A$, i.e., that $3\ne\emptyset$ and $4\ne\emptyset$.

