 Write the negation of the following; i. If she studies, she will pass in examination. ii. If it rains, then they will not go for picnic. iii. If the determine of a system of linear equations is zero then either the system has no solution or it has an infinite number of solutions.

The negation of an "if-then" statement, \( \text{if } P \text{ then } Q \), is logically equivalent to \( P \) and \( \neg Q \) (i.e., the antecedent \( P \) is true, but the consequent \( Q \) is false).

Here are the negations of the given statements:

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### i. **If she studies, she will pass in the examination.**

Negation:  
**She studies, and she does not pass the examination.**

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### ii. **If it rains, then they will not go for a picnic.**

Negation:  
**It rains, and they go for a picnic.**

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### iii. **If the determinant of a system of linear equations is zero, then either the system has no solution or it has an infinite number of solutions.**

Negation:  
**The determinant of the system is zero, and the system has a unique solution.**

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Let me know if you need further clarification!  
Here are **five related questions** to deepen your understanding:  

1. What is the general structure of an "if-then" statement in logic, and how do you negate it?  
2. How does the negation of a compound statement with "or" change its truth value?  
3. What is the importance of understanding logical negations in mathematical proofs?  
4. Can you provide real-world examples where negating conditional statements is useful?  
5. How does the negation of "if-then" statements apply to understanding the contrapositive?

### Tip:  
When negating statements, break them into their simplest components and use logical operators (like "and" and "not") to maintain precision.

Write the negation of the following; i. If she studies, she will pass in examination. ii. If it rains, then they will not go for picnic. iii. If the determinant of a system of linear equations is zero, then either the system has no solution or it has an infinite number of solutions.

Explore the step-by-step negation of conditional statements, such as logical reasoning behind 'if-then' negations and linear system determinants. Ideal for Grades 10-12 students mastering logical equivalence.

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