I'm having a bit of a problem with producing a venn diagram of this relationship. I have three circles: $U$, $V$, $W$. The identity I have to create is:
$$( U \setminus V ) \setminus W = U \setminus ( V \cup W)$$
I thought the shaded region would be the parts that are only in $U$, but that really seems like a wrong answer. Can someone help me out? Cheers
Nope, it's not wrong. Your original instinct was correct.
The definition of set difference is everything in the first set that is not in the second.
Thus the set of elements in U that are not in V, that are also not in W, is equivalent to the set of elements in U that are not in V or W.
$\begin{align} (U\setminus V)\setminus W & = (U\cap V^c) \cap W^c & \text{by } A\setminus B = A\cap B^c \\[1ex] & = U\cap (V^c\cap W^c) & \text {by associativity} \\[1ex] & = U\cap(V\cup W)^c & \text{by DeMorgan's Rule} \\[1ex] & = U\setminus (V\cup W) & \text{by }A\cap B^c = A\setminus B \\[3ex] \therefore (U\setminus V)\setminus W & = U\setminus (V \cup W) \end{align}$
On the Venn diagram, you have three overlapping circles, U,V,W. Show how excluding the to overlap of U with V, and the the overlap of the remainder with W, (the LHS) is the same as excluding from U the overlap with the union of V and W (the RHS).