1. The problem asks which of the numbers $x$, $y$, $z$, or $w$ must equal 0 if $A \cup B = U$. 2. Recall that $A \cup B = U$ means the union of sets $A$ and $B$ covers the entire universal set $U$. 3. Typically, $x$, $y$, $z$, and $w$ represent the sizes of different regions in a Venn diagram of $A$ and $B$ within $U$. 4. If $A \cup B = U$, then there is no element outside both $A$ and $B$. 5. The region outside $A$ and $B$ corresponds to $w$ (assuming $w$ represents elements in $U$ but not in $A$ or $B$). 6. Therefore, $w$ must be 0 because no elements lie outside $A$ or $B$. 7. The other numbers $x$, $y$, and $z$ can be nonzero as they represent elements inside $A$, $B$, or their intersection. Final answer: $w=0$