Using Property Rules to Simplify Set Expressions I've been given a series of 20 set expressions which need to be simplified using property rules as part of an assignment.
I have absolutely no idea what I'm doing. I've researched the subject online but none of the resources I've come across provide examples - which is really what I need before I can understand something.
Question No. 1:
$((Y \cup Y) \cap (X \cap Y)) \cap (X \cap (Y\cap X))$ 
Could somebody kindly provide me with a step by step explanation of how I should go about simplifying such an expression? 

So, the "property rules" are associativity, commutativity, idempotence of $\cap$ and $\cup$, and distributivity of each over the other. Using these, you can simplify your expression as follows (skipping tedious intermediate steps):
$$\begin{align}
((Y \cup Y) \cap (X \cap Y)) \cap (X \cap (Y\cap X)) &= (Y\cup Y)\cap (X\cap Y\cap X\cap Y\cap X) \tag 1 \\
&= Y\cap (X\cap Y) \tag 2 \\
&= X\cap Y \tag 3 \\
\end{align}$$
(1) follows by associativity, (2) and (3) follow from their predecessors by associativity, commutativity and idempotence.
