Question: Which of the numbers $x$, $y$, $z$, or $w$ must equal 0 if $A = B$? The number(s) $x$, $y$, $z$, or $w$ must equal zero if $A = B$. (Use a comma to separate answers as needed.)

1. **State the problem:** We have two sets $A$ and $B$ represented by a Venn diagram with regions labeled $x$, $y$, $z$, and $w$. The values represent the number of elements in each region: - $x$: elements only in $A$ - $y$: elements in both $A$ and $B$ - $z$: elements only in $B$ - $w$: elements outside both $A$ and $B$ We want to find which of $x$, $y$, $z$, or $w$ must be zero if $A = B$. 2. **Recall the meaning of $A = B$:** If $A = B$, then the sets $A$ and $B$ contain exactly the same elements. 3. **Analyze the Venn diagram regions:** - The region $x$ represents elements in $A$ but not in $B$. - The region $z$ represents elements in $B$ but not in $A$. Since $A = B$, there can be no elements exclusively in $A$ or exclusively in $B$. 4. **Conclusion:** - $x$ must be $0$ because no elements are only in $A$. - $z$ must be $0$ because no elements are only in $B$. 5. **About $y$ and $w$:** - $y$ represents elements in both $A$ and $B$, so it can be any number $ eq 0$. - $w$ represents elements outside both $A$ and $B$, so it can be any number. **Final answer:** $$x = 0, z = 0$$