Let `X ={1,2,3,4,5,6,7,8,9}`
`A= {2,4,5,7,8}, B= {1,3,5,7},`
`C={4,6,8,9},` then `A nn (BuuC)`

Options

A. `Ann (BnnC)`
B. `Auu (B nnC)`
C. `(A nnB) uu (A nnC)`
D. `(A uu B) uu (A nn C)`

Answer

To solve the problem step by step, we need to find \( A \cap (B \cup C) \) given the sets:

- \( X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
- \( A = \{2, 4, 5, 7, 8\} \)
- \( B = \{1, 3, 5, 7\} \)
- \( C = \{4, 6, 8, 9\} \)

### Step 1: Find \( B \cup C \)

To find the union of sets \( B \) and \( C \), we combine all the elements from both sets without duplication.

- \( B = \{1, 3, 5, 7\} \)
- \( C = \{4, 6, 8, 9\} \)

Now, we list all unique elements from both sets:

\[ B \cup C = \{1, 3, 4, 5, 6, 7, 8, 9\} \]

### Step 2: Find \( A \cap (B \cup C) \)

Next, we need to find the intersection of set \( A \) with the result from step 1.

- \( A = \{2, 4, 5, 7, 8\} \)
- \( B \cup C = \{1, 3, 4, 5, 6, 7, 8, 9\} \)

Now, we look for common elements in both sets:

- The element \( 4 \) is in both \( A \) and \( B \cup C \).
- The element \( 5 \) is in both \( A \) and \( B \cup C \).
- The element \( 7 \) is in both \( A \) and \( B \cup C \).
- The element \( 8 \) is in both \( A \) and \( B \cup C \).

Thus, the intersection is:

\[ A \cap (B \cup C) = \{4, 5, 7, 8\} \]

### Final Answer

The final result is:

\[ A \cap (B \cup C) = \{4, 5, 7, 8\} \]

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