1. **State the problem:** We have three sets defined on the universal set $\xi = \{x : x \text{ is a positive integer less than } 16\}$. - $I = \{x : x \text{ is a multiple of } 4\}$ - $J = \{x : x \text{ is a factor of } 8\}$ We need to: (i) List all elements of $\xi$, $I$, and $J$ in set notation. (ii) Draw a Venn diagram representing $\xi$, $I$, and $J$. (iii) From the Venn diagram, find: (a) $(I \cup J)'$ (b) $I \cap J'$ --- 2. **List the elements:** - $\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}$ - $I = \{4, 8, 12\}$ since these are multiples of 4 less than 16. - $J = \{1, 2, 4, 8\}$ since these are factors of 8. --- 3. **Understand the sets and their union and complements:** - $I \cup J = \{1, 2, 4, 8, 12\}$ (all elements in either $I$ or $J$ or both). - The complement $(I \cup J)'$ is all elements in $\xi$ not in $I \cup J$. - $J' = \xi \setminus J = \{3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15\}$. - $I \cap J'$ is the intersection of $I$ and $J'$. --- 4. **Calculate $(I \cup J)'$:** $$ (I \cup J)' = \xi \setminus (I \cup J) = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\} \setminus \{1, 2, 4, 8, 12\} = \{3, 5, 6, 7, 9, 10, 11, 13, 14, 15\} $$ --- 5. **Calculate $I \cap J'$:** $$ I \cap J' = \{4, 8, 12\} \cap \{3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15\} = \{12\} $$ --- **Final answers:** - $(I \cup J)' = \{3, 5, 6, 7, 9, 10, 11, 13, 14, 15\}$ - $I \cap J' = \{12\}$