1. **Problem statement:** Given sets $X = \{2, 3\}$, $Y = \{3, 4\}$, and $Z = \{z, y, m\}$ (assuming $Z$ is a set with elements $z, y, m$), find: 1) $Z \times (X \cap Y)$ 2) $(Z - Y) \times X$ 2. **Recall set operations:** - Intersection $X \cap Y$ is the set of elements common to both $X$ and $Y$. - Set difference $Z - Y$ is the set of elements in $Z$ that are not in $Y$. - Cartesian product $A \times B$ is the set of ordered pairs $(a,b)$ where $a \in A$ and $b \in B$. 3. **Calculate $X \cap Y$:** $X = \{2, 3\}$ and $Y = \{3, 4\}$ Common elements: $3$ So, $X \cap Y = \{3\}$ 4. **Calculate $Z \times (X \cap Y)$:** $Z = \{z, y, m\}$ and $X \cap Y = \{3\}$ Cartesian product: $$Z \times (X \cap Y) = \{(z,3), (y,3), (m,3)\}$$ 5. **Calculate $Z - Y$:** $Z = \{z, y, m\}$ and $Y = \{3, 4\}$ Since none of $z, y, m$ are in $Y$, $$Z - Y = \{z, y, m\}$$ 6. **Calculate $(Z - Y) \times X$:** $Z - Y = \{z, y, m\}$ and $X = \{2, 3\}$ Cartesian product: $$ (Z - Y) \times X = \{(z,2), (z,3), (y,2), (y,3), (m,2), (m,3)\} $$ **Final answers:** 1) $Z \times (X \cap Y) = \{(z,3), (y,3), (m,3)\}$ 2) $(Z - Y) \times X = \{(z,2), (z,3), (y,2), (y,3), (m,2), (m,3)\}$