We are tasked with finding $A \cup (B \cap C)$, where the sets are defined as follows:

• $U = \{1, 2, 3, \dots, 8\}$
• $A = \{1, 2, 4, 5\}$
• $B = \{1, 3, 6\}$
• $C = \{1, 2, 3, 4, 5\}$

Step 1: Find $B \cap C$ (the intersection of sets $B$ and $C$)

The intersection of two sets contains all elements that are common to both sets.

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

The common elements between $B$ and $C$ are $1$ and $3$. Therefore: $B \cap C = \{1, 3\}$

Step 2: Find $A \cup (B \cap C)$ (the union of set $A$ and $B \cap C$)

The union of two sets contains all elements that are in either set, without duplicates.

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

Now, take the union of these two sets: $A \cup (B \cap C) = \{1, 2, 4, 5\} \cup \{1, 3\} = \{1, 2, 3, 4, 5\}$

Final Answer:

$A \cup (B \cap C) = \{1, 2, 3, 4, 5\}$

Would you like more details or have any questions?

Related Questions:

1. What is the result of $B \cup C$?
2. What is $A \cap B$?
3. How do we find the complement of set $A$ with respect to $U$?
4. What is $(A \cup B) \cap C$?
5. How do we prove that the union operation is commutative?

Tip:

When dealing with unions and intersections, it's often useful to draw a Venn diagram to visualize the relationships between the sets.