If a:b=c:d=e:f then shows that (a4+a2e2-e4)/(b6+b2f2-f5) = a4/b4

Step 1: Find the union of sets $A$ and $B$, $A \cup B$. $A \cup B = \{1,2,5,6,7\} \cup \{1,3,4,5,6,8\}$ $A \cup B = \{1,2,3,4,5,6,7,8\}$

Step 2: Find the complement of $A \cup B$, denoted as $(A \cup B)'$. This includes all elements in the universal set $U$ that are not in $A \cup B$. $U = \{1,2,3,4,5,6,7,8,9\}$ $(A \cup B)' = U - (A \cup B)$ $(A \cup B)' = \{1,2,3,4,5,6,7,8,9\} - \{1,2,3,4,5,6,7,8\}$ $(A \cup B)' = \{9\}$

Step 3: Find the complement of set $A$, denoted as $A'$. This includes all elements in $U$ that are not in $A$. $A' = U - A$ $A' = \{1,2,3,4,5,6,7,8,9\} - \{1,2,5,6,7\}$ $A' = \{3,4,8,9\}$

Step 4: Find the complement of set $B$, denoted as $B'$. This includes all elements in $U$ that are not in $B$. $B' = U - B$ $B' = \{1,2,3,4,5,6,7,8,9\} - \{1,3,4,5,6,8\}$ $B' = \{2,7,9\}$

Step 5: Find the intersection of $A'$ and $B'$, denoted as $A' \cap B'$. This includes all elements common to both $A'$ and $B'$. $A' \cap B' = \{3,4,8,9\} \cap \{2,7,9\}$ $A' \cap B' = \{9\}$

Step 6: Compare the results from Step 2 and Step 5. We found $(A \cup B)' = \{9\}$ and $A' \cap B' = \{9\}$. Since both sides are equal, the statement is proven. $\boxed{(A \cup B)' = A' \cap B'}$