If `X=x:x gt 0, x^2 lt 0}`, and Y= {flower, Churchill, moon, Kargil), then which one of the following is a correct statement ?

Options

A. X is well defined but Y is not a well defined set
B. Y is well defined but X is not a well defined set
C. Both X and Y are well defined sets
D. Neither X nor Y is a well defined set

Answer

To solve the problem, we need to analyze the two sets given:

1. Set \( X = \{ x : x > 0, x^2 < 0 \} \)
2. Set \( Y = \{ \text{flower, Churchill, moon, Kargil} \} \)

### Step 1: Analyze Set \( X \)

- The condition for set \( X \) states that \( x \) must be greater than 0 and \( x^2 \) must be less than 0.
- Since \( x > 0 \), the possible values of \( x \) could be any positive number (e.g., 1, 2, 3, ...).
- However, the condition \( x^2 < 0 \) is problematic. The square of any real number (whether positive, negative, or zero) is always non-negative. Therefore, \( x^2 \) can never be less than 0.

**Conclusion for Set \( X \)**: Since there are no values of \( x \) that satisfy both conditions, set \( X \) is an empty set (null set).

### Step 2: Determine if Set \( X \) is Well-Defined

- An empty set is considered a well-defined set in set theory. Therefore, set \( X \) is well-defined, even though it contains no elements.

### Step 3: Analyze Set \( Y \)

- Set \( Y \) consists of the elements: flower, Churchill, moon, and Kargil.
- All of these elements are clearly defined and listed.

**Conclusion for Set \( Y \)**: Set \( Y \) is well-defined since it contains specific, identifiable elements.

### Step 4: Conclusion

- Both sets \( X \) and \( Y \) are well-defined.
- Therefore, the correct statement is that both \( X \) and \( Y \) are well-defined sets.

### Final Answer

The correct option is: **Both \( X \) and \( Y \) are well-defined.**

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