Can you help me with this?
Let $x=A \mid B$ and x'=A' \mid B' be cuts in $\mathbb Q$. It is defined
x+x'=(A+A') \mid \text{ rest of } \mathbb Q
Show that although B+B' is disjoint from A+A', it may happen in degenerate cases that $\mathbb Q$ is not the union of A+A' and B+B'.
EDIT: As the comment below asked, I'll include the definition of a cut in $\mathbb Q$.
A cut in $\mathbb Q$ is a pair of subsets $A,B$ of $\mathbb Q$ such that
(a) $A\cup B=\mathbb Q$, $A\neq \emptyset$, $B\neq \emptyset$, $A\cap B=\emptyset$
(b) if $a\in A$ and $b\in B$ then $a.
(c) $A$ contains no largest element.