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.