Given that A = {1, 5, 7} B = {3, 9, 12, 15} C = {2, 4, 6, 8} Find (A ∪ B) ∪ C

1. Analyze the question

The question asks for the union of three sets: $A$, $B$, and $C$. The union symbol ($\cup$) means we need to combine all the elements from the given sets into one single set, ensuring that no element is repeated.

2. Explain the concept

The union of sets $A, B,$ and $C$ (denoted as $A \cup B \cup C$) is the set of all elements that are members of $A$, or $B$, or $C$.

• Rule: List every unique element found in any of the sets.
• Strategy: It is often easiest to list them in ascending order to ensure you don't miss any numbers and to make the final set look organized.

3. Show the solution

Given:

• $A = \{1, 5, 7\}$
• $B = \{3, 9, 12, 15\}$
• $C = \{2, 4, 6, 8\}$

Step 1: Find $(A \cup B)$ Combine all elements from $A$ and $B$: $A \cup B = \{1, 3, 5, 7, 9, 12, 15\}$

Step 2: Find $(A \cup B) \cup C$ Now, take the result from Step 1 and add all elements from set $C$: $(A \cup B) \cup C = \{1, 3, 5, 7, 9, 12, 15\} \cup \{2, 4, 6, 8\}$

Step 3: Arrange in ascending order Combine and sort the numbers: $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15\}$

4. Identify the answer

Comparing our result to the options provided: A. {1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15} B. {1, 2, 3, 5, 6, 8, 12, 15} C. {2, 4, 5, 9, 12, 15} D. {1, 5, 6, 7, 8, 9, 12, 15}

The calculated set matches option A.

5. Tip

When solving union problems, always cross out the numbers in the original sets as you write them down in your final set. This prevents you from accidentally skipping a number or writing one twice. Also, remember that the order of elements in a set does not matter, but writing them in ascending order makes it much easier to compare your answer with the options!