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The given statement is:

**"Square of a negative real number is positive."**

We need to negate this statement, and we have the following options.

Let's break down the original statement:
1. **Original statement in logical terms:** "For all \( x \), if \( x \) is less than 0, then \( x^2 > 0 \)" (since the square of any real number is positive).
   - In symbolic form: \( \forall x ((x < 0) \to (x^2 > 0)) \).

### Negation of the statement:
The negation of the statement "For all \( x \), if \( x < 0 \), then \( x^2 > 0 \)" would be "There exists some \( x \) such that \( x < 0 \) and \( x^2 \leq 0 \)" (i.e., the square of some negative number is non-positive, meaning it is 0 or negative).

This leads us to the option that matches the negation:

- **Option C:** \( \exists x ((x < 0) \land (x^2 \leq 0)) \)

This correctly represents the negation of the original statement. Therefore, the correct answer is:

**C)** \( \exists x ((x < 0) \land (x^2 \leq 0)) \).

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Would you like more details on the logical steps involved? Here are five related questions you can explore:

1. What are the rules for negating "for all" (\( \forall \)) and "there exists" (\( \exists \)) quantifiers?
2. How do implications (\( \to \)) work in logical statements?
3. Why does the square of a real number always result in a non-negative number?
4. What are the properties of real numbers under squaring?
5. How do logical conjunction (\( \land \)) and disjunction (\( \lor \)) work in truth tables?

**Tip:** When negating a statement involving "for all" (\( \forall \)), it turns into "there exists" (\( \exists \)), and the internal logical condition flips accordingly.

Which of the following is the negation of the statement 'Square of a negative real number is positive'?

Learn how to find the negation of a statement involving quantifiers and real numbers, specifically 'Square of a negative real number is positive.' This problem involves key concepts in logic, negation, and quantifiers. Suitable for advanced high school or undergraduate students.

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