Let's break this down step by step:

Step 1: Understand the sets.

We are given the following sets:

• $U = \{1, 2, 3, \dots, 6\}$
• $A = \{1, 2, 3, 4\}$
• $B = \{1, 2, 3\}$
• $C = \{1, 2, 3, 4, 5\}$

Step 2: Find $B \cap C$.

The intersection of sets $B$ and $C$ includes all the elements that are common to both sets:

• $B = \{1, 2, 3\}$
• $C = \{1, 2, 3, 4, 5\}$

So, $B \cap C = \{1, 2, 3\}$.

Step 3: Find $A \cup (B \cap C)$.

Now we need to find the union of set $A$ with $B \cap C$:

• $A = \{1, 2, 3, 4\}$
• $B \cap C = \{1, 2, 3\}$

The union of these two sets includes all the elements from both sets, without duplicates:

• $A \cup (B \cap C) = \{1, 2, 3, 4\}$

Final Answer:

$A \cup (B \cap C) = \{1, 2, 3, 4\}$

Would you like further details or have any questions?

Here are 5 related questions:

1. How is the intersection of two sets different from their union?
2. What would be the result of $B \cup C$?
3. Can you calculate $A \cap B$?
4. How would the union change if $A$ had more elements than $B \cap C$?
5. What is the difference between complement and intersection in set theory?

Tip: The union of two sets contains all unique elements from both sets, while the intersection contains only elements common to both sets.