Step 1 :Given the sets A = {1, 2, 3, 4}, B = {7, 8, 9}, and C = {1, 2, 4, 5, 6}.
Step 2 :First, we need to find the intersection of sets B and C, denoted as \(B \cap C\). The intersection of two sets is a set containing all elements that are common to both sets.
Step 3 :Sets B and C have no common elements, so the intersection of sets B and C is an empty set.
Step 4 :Next, we find the union of set A and the result of \(B \cap C\), denoted as \(A \cup(B \cap C)\). The union of two sets is a set containing all elements that are in either set.
Step 5 :The union of set A and an empty set is set A itself.
Step 6 :Therefore, the set \(A \cup(B \cap C)\) is equal to set A.
Step 7 :Final Answer: \(A \cup(B \cap C)=\boxed{\{1,2,3,4\}}\)