Q:

Expressions for $(AB)^R$, $(A \cap B)^R$, $(A \cup B)^R$

For any language A, let $A^R$ be $\{x^R \mid x \in A\}$. Then, for arbitrary languages $A$ and $B$, explicitly write down the expressions for $(AB)^R$, $(A \cap B)^R$, $(A \cup B)^R$.
I am not really sure what this question is asking me to solve.  I understand that $A^R$ is the reversal of a set, but what does it mean in terms of a language?  What does it mean for the union of $A$ and $B$?  or the intersection of $A$ and $B$?
I have defined the set $A = \{ad, cd\}$ and $B = \{bc, ab\}$, so the reversal of the set $A$ is $\{da, dc\}$.  Can I do this?  Is this a language? Should I chose $A$ and $B$ to be different things, as I don't know how to can show a union or intersection ?  
Any help would be greatly appreciated!

A:

You first have to make sure you fully understand the definitions you are using.

Reversal of a word: if $u = a_1a_2 \cdots a_n$ is a word, then its reversal $u^R$ is $a_n \cdots a_2a_1$.
Language. If $X$ is the alphabet, then a language is simply a subset of $X^*$.
Reversal of a language. If $L$ is a language, then its reversal $L^R$ is the set of all reversals $u^R$ such that $u \in L$.
Union. $A \cup B = \{ u \in X^* \mid u \in A \text{ or } u \in B\}$.
Intersection. $A \cap B = \{ u \in X^* \mid u  \in A \text{ and } u \in B\}$.
Concatenation.  $AB = \{ w \in X^* \mid w = uv \text{ for some } u \in A \text{ and } v \in B\}$.

Now, it suffices to apply the definitions step by step. I show you for the reversal of the union.
The reversal of $A \cup B$ is the set of all words $u^R$ such that $u \in A \cup B$. By (4), this is also the set of all words $u^R$ such that $u \in A$ or $u \in B$. By (3), the set of all words $u^R$ such that $u \in A$ is $A^R$ and the set of all words $u^R$ such that $u \in B$ is $B^R$. Consequently $(A \cup B)^R = A^R \cup B^R$.
The reversal of the intersection is also easy, but the reversal of $AB$ is a little bit more tricky. You should first understand what is $(uv)^R$ when $u$ and $v$ are words.

