Let's break down the given steps and solve each part systematically:

Part a) Find the elements in the set $P \cap Q \cap R$

1. Compare $P$ and $Q$:

   • $P = \{3, 6, 9, 11\}$
   • $Q = \{2, 3, 4, 5, 6\}$
   • Common elements between $P$ and $Q$: $\{3, 6\}$

2. Compare the result with $R$:

   • $R = \{1, 4, 5, 6, 8\}$
   • Common elements between $P \cap Q = \{3, 6\}$ and $R = \{1, 4, 5, 6, 8\}$: $\{6\}$

So, the intersection of all three sets $P \cap Q \cap R$ is:

$P \cap Q \cap R = \{6\}$

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Part b) Find the elements in set $P \cup (Q \cap R)$ and represent it visually

1. Find the intersection of $Q$ and $R$:

   • $Q = \{2, 3, 4, 5, 6\}$
   • $R = \{1, 4, 5, 6, 8\}$
   • Common elements between $Q$ and $R$: $\{4, 5, 6\}$
   • Therefore, $Q \cap R = \{4, 5, 6\}$

2. Find the union of $P$ and $(Q \cap R)$:

   • $P = \{3, 6, 9, 11\}$
   • $Q \cap R = \{4, 5, 6\}$
   • Combine all elements from $P$ and $Q \cap R$:

   $P \cup (Q \cap R) = \{3, 6, 9, 11\} \cup \{4, 5, 6\} = \{3, 4, 5, 6, 9, 11\}$

So, the union of $P$ and $(Q \cap R)$ is:

$P \cup (Q \cap R) = \{3, 4, 5, 6, 9, 11\}$

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Venn Diagram Representation

To represent this visually using a Venn diagram, we would draw three overlapping circles corresponding to the sets $P$, $Q$, and $R$, and then shade the region corresponding to $P \cup (Q \cap R)$.

• The union $P \cup (Q \cap R)$ will include:
  • All elements in $P$ (i.e., $\{3, 6, 9, 11\}$)
  • All elements in the intersection $Q \cap R$ (i.e., $\{4, 5, 6\}$)
  • Therefore, the shaded region should include: $\{3, 4, 5, 6, 9, 11\}$, covering parts of all three sets.

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Would you like a more detailed explanation or a drawn Venn diagram? Feel free to ask if you have any questions!

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5 Follow-up Questions:

1. What is the difference between the union and intersection of sets?
2. How would you visualize the union of two sets in a Venn diagram?
3. How do you calculate the difference between two sets?
4. How does the complement of a set relate to Venn diagrams?
5. How do you approach solving set problems involving multiple operations (like union, intersection, etc.)?

Tip: When dealing with Venn diagrams, always make sure to first identify all the sets involved and what operations are required (e.g., union, intersection, etc.), then visualize their relationships step-by-step.