1. **State the problem:** We need to shade the sets $X \cap (Y \cup Z)$ and $(X \cap Y) \cup (X \cap Z)$ in Venn diagrams and then conclude the relationship between these two sets. 2. **Recall set theory formulas:** - $X \cap (Y \cup Z)$ means the elements that are in $X$ and also in either $Y$ or $Z$ (or both). - $(X \cap Y) \cup (X \cap Z)$ means the elements that are either in both $X$ and $Y$, or in both $X$ and $Z$. 3. **Show the equivalence:** By distributive law of sets, $$ X \cap (Y \cup Z) = (X \cap Y) \cup (X \cap Z) $$ This means these two sets are equal. 4. **Explain the shading:** - For $X \cap (Y \cup Z)$, shade all parts of $X$ that overlap with $Y$ or $Z$, including the triple overlap. - For $(X \cap Y) \cup (X \cap Z)$, shade the overlap of $X$ and $Y$ combined with the overlap of $X$ and $Z$, including the triple overlap. 5. **Conclusion:** The two sets $X \cap (Y \cup Z)$ and $(X \cap Y) \cup (X \cap Z)$ are equal, so their shaded regions in the Venn diagrams are identical. This demonstrates the distributive property of intersection over union in set theory.