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

Answer

To determine the truth value of the statement "For all \( x \) in \( A \), \( x \) is an even number," we will follow these steps:

### Step 1: Define the Set \( A \)
The set \( A \) is given as:
\[ A = \{ 2, 4, 6, 8, 10 \} \]

### Step 2: Understand the Statement
The statement we need to evaluate is:
"For all \( x \) in \( A \), \( x \) is an even number."

This means we need to check if every element \( x \) in the set \( A \) satisfies the condition of being an even number.

### Step 3: Check Each Element in \( A \)
We will check each element of the set \( A \) to see if it is even:

1. **Check \( x = 2 \)**:
- \( 2 \div 2 = 1 \) (no remainder) → Even
2. **Check \( x = 4 \)**:
- \( 4 \div 2 = 2 \) (no remainder) → Even
3. **Check \( x = 6 \)**:
- \( 6 \div 2 = 3 \) (no remainder) → Even
4. **Check \( x = 8 \)**:
- \( 8 \div 2 = 4 \) (no remainder) → Even
5. **Check \( x = 10 \)**:
- \( 10 \div 2 = 5 \) (no remainder) → Even

### Step 4: Conclusion
Since all elements \( 2, 4, 6, 8, \) and \( 10 \) in the set \( A \) are even numbers, we conclude that the statement "For all \( x \) in \( A \), \( x \) is an even number" is **true**.

### Final Answer
The truth value of the statement is **True**.

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