Given the following​ sets, find the set Upper A union left parenthesis Upper B intersect Upper C right parenthesis.
Uequals​{​1, ​2, 3, . .​ ., 8​}
Aequals​{1​, 2​, 4​, 5​}
Bequals​{1​, 3​, 6​}
Cequals​{1​, 2​, 3​, 4​, 5​}

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.

Given the following​ sets, find the set Upper A union left parenthesis Upper B intersect Upper C right parenthesis. Uequals​{​1, ​2, 3, . .​ ., 8​}, Aequals​{1​, 2​, 4​, 5​}, Bequals​{1​, 3​, 6​}, Cequals​{1​, 2​, 3​, 4​, 5​}.

Learn how to solve the set theory problem where you find A ∪ (B ∩ C) for sets A = {1, 2, 4, 5}, B = {1, 3, 6}, and C = {1, 2, 3, 4, 5}. This problem covers key set theory concepts such as union and intersection. Suitable for students in Grades 8-10.

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