Let A ={ 2,4,6,8,10} . Determine the truth value of each of the following :
`EE x in A,x`is a prime number.

Answer

To determine the truth value of the statement "∃! x ∈ A, x is a prime number," we need to analyze the set A and identify the prime numbers within it.

### Step-by-Step Solution:

1. **Identify the Set A**:
The set A is given as \( A = \{ 2, 4, 6, 8, 10 \} \).

2. **Define Prime Numbers**:
A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself.

3. **Check Each Element in A**:
- **2**:
- Divisors: 1, 2
- Conclusion: 2 is a prime number.
- **4**:
- Divisors: 1, 2, 4
- Conclusion: 4 is not a prime number.
- **6**:
- Divisors: 1, 2, 3, 6
- Conclusion: 6 is not a prime number.
- **8**:
- Divisors: 1, 2, 4, 8
- Conclusion: 8 is not a prime number.
- **10**:
- Divisors: 1, 2, 5, 10
- Conclusion: 10 is not a prime number.

4. **Count the Prime Numbers in A**:
From the analysis, we find that the only prime number in the set A is 2. Thus, there is exactly one prime number in the set A.

5. **Evaluate the Statement**:
The statement "∃! x ∈ A, x is a prime number" translates to "There exists exactly one x in A such that x is a prime number." Since we found that there is indeed exactly one prime number (which is 2), the statement is true.

### Conclusion:
The truth value of the statement "∃! x ∈ A, x is a prime number" is **True**.

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