1. **State the problem:** We need to construct the intersection graph of the collection of sets $A_1, A_2, A_3, A_4, A_5$ where each vertex represents a set and an edge exists between two vertices if their corresponding sets have at least one common element. 2. **List the sets:** - $A_1 = \{0, 2, 6, 8\}$ - $A_2 = \{0, 1, 2, 3, 4\}$ - $A_3 = \{1, 3, 5, 7, 9\}$ - $A_4 = \{5, 6, 7, 8, 9\}$ - $A_5 = \{0, 1, 8, 9\}$ 3. **Check intersections between each pair:** - $A_1 \cap A_2 = \{0, 2\} \neq \emptyset$ so edge between $A_1$ and $A_2$ - $A_1 \cap A_3 = \emptyset$ no edge - $A_1 \cap A_4 = \{6, 8\} \neq \emptyset$ edge between $A_1$ and $A_4$ - $A_1 \cap A_5 = \{0, 8\} \neq \emptyset$ edge between $A_1$ and $A_5$ - $A_2 \cap A_3 = \{1, 3\} \neq \emptyset$ edge between $A_2$ and $A_3$ - $A_2 \cap A_4 = \emptyset$ no edge - $A_2 \cap A_5 = \{0, 1\} \neq \emptyset$ edge between $A_2$ and $A_5$ - $A_3 \cap A_4 = \{5, 7, 9\} \neq \emptyset$ edge between $A_3$ and $A_4$ - $A_3 \cap A_5 = \{1, 9\} \neq \emptyset$ edge between $A_3$ and $A_5$ - $A_4 \cap A_5 = \{8, 9\} \neq \emptyset$ edge between $A_4$ and $A_5$ 4. **Summary of edges:** - $A_1$ connected to $A_2, A_4, A_5$ - $A_2$ connected to $A_1, A_3, A_5$ - $A_3$ connected to $A_2, A_4, A_5$ - $A_4$ connected to $A_1, A_3, A_5$ - $A_5$ connected to $A_1, A_2, A_3, A_4$ 5. **Final answer:** The intersection graph has vertices $\{A_1, A_2, A_3, A_4, A_5\}$ with edges: $$ \{(A_1,A_2), (A_1,A_4), (A_1,A_5), (A_2,A_3), (A_2,A_5), (A_3,A_4), (A_3,A_5), (A_4,A_5)\} $$ This graph shows all pairs of sets that share at least one common element.