Problem
Let $S$ be the universal set, where: \[ S=\{1,2,3, \ldots, 28,29,30\} \] Let sets $A$ and $B$ be subsets of $S$, where: Set $A=\{1,2,4,6,9,12,16,23,24,29\}$ Set $B=\{2,5,10,12\}$ Set $C=\{1,11,12,15,17,18,22,28,30\}$ LIST the elements in the set $(A \cup B \cup C)$ \[ (A \cup B \cup C)=\{ \] Enter the elements as a list, separated by commas. If the result is the empty set, enter DNE LIST the elements in the set $(A \cap B \cap C)$ \[ (A \cap B \cap C)=\{ \] Enter the elements as a list, separated by commas. If the result is the empty set, enter DNE You may want to draw a Venn Diagram to help answer this question.
Solution
Step 1 ：Define the sets A, B, and C as follows: \(A=\{1,2,4,6,9,12,16,23,24,29\}\), \(B=\{2,5,10,12\}\), and \(C=\{1,11,12,15,17,18,22,28,30\}\)
Step 2 ：To find the union of sets A, B, and C, denoted as \(A \cup B \cup C\), list all the elements in sets A, B, and C, without repeating any elements
Step 3 ：So, \(A \cup B \cup C=\{1,2,4,6,9,12,16,23,24,29,5,10,11,15,17,18,22,28,30\}\)
Step 4 ：To find the intersection of sets A, B, and C, denoted as \(A \cap B \cap C\), list all the elements that are common to sets A, B, and C
Step 5 ：Looking at sets A, B, and C, the only common element is 12
Step 6 ：So, \(A \cap B \cap C=\boxed{\{12\}}\)
