If `X={1,2,3,4,5,6,7,8,9},A={x inN:30ltx^(2)lt70}`, B = {x x is a prime number less thean 10}, then which of the following is false?

Options

A. `AuuB={2,3,5,6,7,8}`
B. `AnnB={7,8}`
C. `A-B={6,8}`
D. `ADeltaB={2,3,5,6,8}`

Answer

To solve the problem step by step, we need to evaluate the sets A and B based on the definitions provided and then determine which statement about these sets is false.

### Step 1: Define the Sets
- **Set X**: \( X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
- **Set A**: \( A = \{ x \in \mathbb{N} : 30 < x^2 < 70 \} \)
- **Set B**: \( B = \{ x \in \mathbb{N} : x \text{ is a prime number less than 10} \} \)

### Step 2: Determine Set A
To find the elements of set A, we need to find natural numbers \( x \) such that \( 30 < x^2 < 70 \).

- Calculate the square roots:
- \( \sqrt{30} \approx 5.477 \) (so \( x \) must be greater than 5)
- \( \sqrt{70} \approx 8.366 \) (so \( x \) must be less than 8)

Thus, the possible values for \( x \) are \( 6, 7, 8 \) since these are the only natural numbers between \( 5.477 \) and \( 8.366 \).

So, \( A = \{6, 7, 8\} \).

### Step 3: Determine Set B
Set B consists of prime numbers less than 10. The prime numbers in this range are:
- \( 2, 3, 5, 7 \)

So, \( B = \{2, 3, 5, 7\} \).

### Step 4: Evaluate A Union B
Now we find the union of sets A and B:
- \( A \cup B = \{6, 7, 8\} \cup \{2, 3, 5, 7\} = \{2, 3, 5, 6, 7, 8\} \)

### Step 5: Evaluate A Intersection B
Next, we find the intersection of sets A and B:
- \( A \cap B = \{6, 7, 8\} \cap \{2, 3, 5, 7\} = \{7\} \)

### Step 6: Evaluate A Minus B
Now we find the difference of A and B:
- \( A - B = \{6, 7, 8\} - \{2, 3, 5, 7\} = \{6, 8\} \)

### Step 7: Evaluate A Delta B (Symmetric Difference)
Finally, we find the symmetric difference:
- \( A \Delta B = (A \cup B) - (A \cap B) = \{2, 3, 5, 6, 7, 8\} - \{7\} = \{2, 3, 5, 6, 8\} \)

### Step 8: Conclusion
Now we can evaluate the options given in the question to identify which statement is false. Based on our evaluations:
- \( A \cup B = \{2, 3, 5, 6, 7, 8\} \) (True)
- \( A \cap B = \{7\} \) (True)
- \( A - B = \{6, 8\} \) (True)
- \( A \Delta B = \{2, 3, 5, 6, 8\} \) (True)

The only false statement would be one that incorrectly states the intersection or any other evaluated set.