1. **Problem Statement:** 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 edges connect vertices whose sets have non-empty intersections. 2. **Sets Given:** - $A_1 = \{0, 2, 6, 8\}$ - $A_2 = \{0, 1, 2, 3, 4\}$ - $A_3 = \{1, 3, 5, 7, 9\}$ - $A_4 = \{5, 6, 7, 8, 9\}$ - $A_5 = \{0, 1, 8, 9\}$ 3. **Rule for Edges:** An edge exists between vertices $A_i$ and $A_j$ if $A_i \cap A_j \neq \emptyset$. 4. **Find Intersections:** - $A_1 \cap A_2 = \{0, 2\}$ (non-empty) \Rightarrow$ edge between $A_1$ and $A_2$ - $A_1 \cap A_3 = \emptyset$ (no edge) - $A_1 \cap A_4 = \{6, 8\}$ (non-empty) \Rightarrow$ edge between $A_1$ and $A_4$ - $A_1 \cap A_5 = \{0, 8\}$ (non-empty) \Rightarrow$ edge between $A_1$ and $A_5$ - $A_2 \cap A_3 = \{1, 3\}$ (non-empty) \Rightarrow$ edge between $A_2$ and $A_3$ - $A_2 \cap A_4 = \emptyset$ (no edge) - $A_2 \cap A_5 = \{1\}$ (non-empty) \Rightarrow$ edge between $A_2$ and $A_5$ - $A_3 \cap A_4 = \{5, 7, 9\}$ (non-empty) \Rightarrow$ edge between $A_3$ and $A_4$ - $A_3 \cap A_5 = \{1\}$ (non-empty) \Rightarrow$ edge between $A_3$ and $A_5$ - $A_4 \cap A_5 = \{8, 9\}$ (non-empty) \Rightarrow$ edge between $A_4$ and $A_5$ 5. **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 $A_1, A_2, A_3, A_4$ 6. **Final Intersection Graph:** Vertices $A_1, A_2, A_3, A_4, A_5$ with edges as above represent the intersection graph of the given sets. This graph can be visualized as a complete graph minus the edge between $A_2$ and $A_4$ (since they have no common elements).