p : For every integer x, `x^2` is positive integer, (x `ne` 0) statement p can be interpreted as

Options

A. There is at least one integer exists such that its square is positive integer
B. There is at most one integer exists such that its square is positive integer
C. In the set of integers square of all integers are positive integer
D. All of these

Answer

To interpret the statement \( p \): "For every integer \( x \), \( x^2 \) is a positive integer (where \( x \neq 0 \))", we can follow these steps:

### Step 1: Understand the Original Statement
The original statement asserts that for every integer \( x \) (except zero), the square of \( x \) is a positive integer. This means that no matter which integer you choose (as long as it is not zero), when you square it, the result will always be a positive integer.

### Step 2: Analyze the Options
We need to analyze the given options to find an interpretation that maintains the same meaning as the original statement.

1. **Option 1**: "There is at least one integer exists such that its square is a positive integer."
- This option suggests that there is at least one integer with a positive square, which is weaker than the original statement that claims this is true for every integer. Thus, it changes the meaning.

2. **Option 2**: "There is at most one integer exists such that its square is a positive integer."
- This option implies a limitation on the number of integers with positive squares, which again changes the meaning of the original statement. The original statement does not suggest any limitation.

3. **Option 3**: "In the set of integers, the square of all integers is a positive integer."
- This option correctly reflects the original statement's meaning. It asserts that for every integer (excluding zero), squaring it results in a positive integer.

### Step 3: Conclusion
Based on the analysis, the correct interpretation of statement \( p \) is:
**Option 3**: "In the set of integers, the square of all integers is a positive integer."