Let `X = {1,2,3,4,5,6,7,8,9,10}` be the universal set and `A ={2,4,6,} , B = {1,3,7},` then `A^(@) nn B^(@)` is equal to .

Options

A. `{2,4,5,6,7,8,9,10}`
B. `{1,3,5,7,8,9,10}`
C. X
D. `{5,8,9,10}`

Answer

To solve the problem, we need to find the intersection of the complements of sets A and B with respect to the universal set X. Let's break this down step by step.

### Step 1: Identify the Universal Set and the Sets A and B
- The universal set \( X = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
- Set \( A = \{2, 4, 6\} \)
- Set \( B = \{1, 3, 7\} \)

### Step 2: Find the Complement of Set A
The complement of set A, denoted \( A' \) or \( A^(@) \), consists of all the elements in the universal set X that are not in A.

- Elements in A: \( 2, 4, 6 \)
- Therefore, the elements not in A (i.e., in \( A' \)) are:
\[ A' = X - A = \{1, 3, 5, 7, 8, 9, 10\} \]

### Step 3: Find the Complement of Set B
Similarly, the complement of set B, denoted \( B' \) or \( B^(@) \), consists of all the elements in the universal set X that are not in B.

- Elements in B: \( 1, 3, 7 \)
- Therefore, the elements not in B (i.e., in \( B' \)) are:
\[ B' = X - B = \{2, 4, 5, 6, 8, 9, 10\} \]

### Step 4: Find the Intersection of Complements A' and B'
Now we need to find the intersection of the two complements \( A' \) and \( B' \).

- \( A' = \{1, 3, 5, 7, 8, 9, 10\} \)
- \( B' = \{2, 4, 5, 6, 8, 9, 10\} \)

The intersection \( A' \cap B' \) is the set of elements that are common to both \( A' \) and \( B' \):
- Common elements: \( 5, 8, 9, 10 \)

Thus,
\[ A' \cap B' = \{5, 8, 9, 10\} \]

### Final Answer
The value of \( A' \cap B' \) is \( \{5, 8, 9, 10\} \).

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