Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 1 / b / Solution

By definition the symmetric difference is $A\oplus B=(A\setminus B)\cup (B\setminus A)$. Put $D=A\oplus B$ and $E=A\cap B$. Distribution of set intersection over set union givesFor disjoint sets, set difference removes nothing, so $D\oplus E=(D\setminus E)\cup (E\setminus D)=D\cup E$. ConsequentlyHere the second equality combines $A\cap B^c$ with $A\cap B$; the last uses $A\cup A^c=\mathcal{U}$.