`if A={1,2,3,4,5},B={2,4,6}and C={3,4,6}`, then `(Acup B) cap C ` is

Options

A. {3,4,6}
B. {1,2,3}
C. {1,4,3}
D. none of these

Answer

To solve the problem, we need to find \((A \cup B) \cap C\) where \(A = \{1, 2, 3, 4, 5\}\), \(B = \{2, 4, 6\}\), and \(C = \{3, 4, 6\}\).

### Step-by-Step Solution:

**Step 1: Find \(A \cup B\)**
The union of two sets \(A\) and \(B\) includes all the elements from both sets without duplicates.
- \(A = \{1, 2, 3, 4, 5\}\)
- \(B = \{2, 4, 6\}\)

Now, let's combine them:
- From \(A\), we have \(1, 2, 3, 4, 5\)
- From \(B\), we add \(6\) (since \(2\) and \(4\) are already in \(A\))

Thus,
\[ A \cup B = \{1, 2, 3, 4, 5, 6\} \]

**Step 2: Find \((A \cup B) \cap C\)**
Now we need to find the intersection of \(A \cup B\) and \(C\). The intersection includes only those elements that are present in both sets.
- \(A \cup B = \{1, 2, 3, 4, 5, 6\}\)
- \(C = \{3, 4, 6\}\)

Now, we look for common elements:
- The elements in \(A \cup B\) are \(1, 2, 3, 4, 5, 6\)
- The elements in \(C\) are \(3, 4, 6\)

The common elements are \(3\) and \(4\) (since \(6\) is not in \(A \cup B\)). Thus,
\[ (A \cup B) \cap C = \{3, 4\} \]

### Final Answer:
\[ (A \cup B) \cap C = \{3, 4\} \]