https://file.aiquickdraw.com/mathbot/file/1727021013974nkyfozrc.jpg

Let's solve the problem in the image step by step.

### Given:
- \( P = \{4, 6, 12, 14\} \)
- \( Q = \{1, 7, 11\} \)
- \( R = \{4, 7, 8, 11\} \)

We are asked to find \( P \cup (Q \cap R) \).

### Step 1: Find the intersection \( Q \cap R \)
The intersection \( Q \cap R \) is the set of elements that are common to both \( Q \) and \( R \). 

- \( Q = \{1, 7, 11\} \)
- \( R = \{4, 7, 8, 11\} \)

The common elements are \( \{7, 11\} \), so:
\[
Q \cap R = \{7, 11\}
\]

### Step 2: Find the union \( P \cup (Q \cap R) \)
Now we take the union of set \( P \) and the result from step 1, \( Q \cap R \).

- \( P = \{4, 6, 12, 14\} \)
- \( Q \cap R = \{7, 11\} \)

The union combines all unique elements from both sets:
\[
P \cup (Q \cap R) = \{4, 6, 12, 14, 7, 11\}
\]

Thus, the final answer is:
\[
\boxed{4, 6, 7, 11, 12, 14}
\]

Would you like further details or have any other questions? Here are 5 related questions:

1. How do you calculate the difference between two sets, for example, \( P - Q \)?
2. What is the intersection of three sets \( P \), \( Q \), and \( R \)?
3. How would the answer change if we used the union instead of the intersection in \( Q \cup R \)?
4. Can you explain what a complement of a set is and how to calculate it?
5. How would you represent the problem using a Venn diagram?

**Tip:** When solving set operations, always remember that intersections include common elements, while unions combine all elements.

For P = {4, 6, 12, 14}, Q = {1, 7, 11}, and R = {4, 7, 8, 11}, find P ∪ (Q ∩ R).

Learn how to solve the set theory problem involving union and intersection: P ∪ (Q ∩ R). The problem requires finding the common elements between sets Q and R, followed by the union with set P. Detailed steps included for Grades 8-10.

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