Proving that a relation R is an equivalence relation While I fully understand what it means to be an equivalence relation, I have a difficulty establishing proof that $R$ is an equivalence relation without just listing all pairs that $R$ creates and testing them.
However this method is greatly time consuming and is not possible during exams as we usually have only 2 minutes (exam is 120 min long and is out of 120 marks, the question below is worth 2 marks only) to show that R is an equivalence relation.
For the following relation, can someone show me a fast method for proving that $R$ is an equivalence relation?

Let $\mathcal{P}(S)$ be the power set of $S =\{0,1,2,...,9\}$ and define $$R = \{ (A,B) \in \mathcal{P}(S) \times \mathcal{P}(S) : A=S\backslash B \text{ or } A=B\}.$$

I know we can use $A=B$ from the relation definition to assert that it is reflexive, but what about symmetry and transitivity?
If I prove that $xRy$ is the same as $yRx$ for one example, that doesn't prove that all $A$s and $B$s have symmetric relations as there might be a contradiction somewhere, or is just proving one example symmetric enough to assert that all $A$s and $B$s have a symmetric relation?

$A$ and $B$ are related iff they are equal or complements. 
Reflexivity: For every subset $A$  of $S$ we have $A=A$
Symmetry: If $A$ is related to $B$ then either they are equal or complements, so $B$ is also related to $A$
Transitivity: If $A$ is related to $B$ and $B$ is related to $C$, then either $B=A$ or $B$ is complement of $A$ and either $C=B$ or $C$ is complement of $B$
In either case we have $A=C$ or $C$ is complement of $A$ 
