GSEB Class 11 Maths Solutions Chapter 14 Mathematical Reasoning Exercise 14.4

Question 1. Rewrite the following statement with 'if-then' in five different ways, conveying the same meaning: 'If a natural number is odd, then its square is also odd'.
Answer:
1. A natural number being odd suggests that its square will also be odd.
2. The natural number is odd only if its square also turns out to be odd.
3. If the square of a natural number is not odd, then that natural number is also not odd.
4. For a natural number to be odd, it is necessary that its square also be odd.
5. For a square of a natural number to be odd, it is sufficient that the number itself is odd.
In simple words: We can say the same idea in five different ways: An odd number's square is odd. This is true if the square is odd. If the square is not odd, the number isn't either. The square must be odd for the number to be odd. The number being odd is enough for its square to be odd.

Exam Tip: When rephrasing statements in 'if-then' form, ensure each variation logically conveys the exact same relationship and implication as the original statement, using terms like 'implies', 'only if', 'necessary', and 'sufficient'.

Question 2. Write the contrapositives and converses of the following statements:
1. If x is a prime number, then x is odd.
2. If the two lines are parallel, then they do not intersect in the same plane.
3. Something is cold implies that it has low temperature.
4. You cannot comprehend geometry, if you do not know how to reason deductively.
5. x is an even number implies that x is divisible by 4.
Answer:
1. Contrapositive statement: If a number x is not odd, then x is not a prime number.
Converse statement: If x is odd, then x is a prime number.

2. Contrapositive statement: If two lines intersect in a plane, then the lines are not parallel.
Converse statement: If two lines do not intersect in the same plane, then the two lines are parallel.

3. Contrapositive statement: If the temperature of something is not low, then it is not cold.
Converse statement: If something has low temperature, then it is cold.

4. Contrapositive statement: If you can comprehend geometry, then you know how to reason deductively.
Converse statement: If you do not know how to reason deductively, then you cannot comprehend geometry.

5. Contrapositive statement: If x is not divisible by 4, then x is not an even number.
Converse statement: If x is divisible by 4, then x is an even number.
In simple words: To find the contrapositive, swap the "if" and "then" parts and make both negative. To find the converse, just swap the "if" and "then" parts without making them negative. Remember to change the original statement's parts when you swap them for the contrapositive.

Exam Tip: For conditional statements "If P, then Q": the converse is "If Q, then P", and the contrapositive is "If not Q, then not P". Understanding these transformations is crucial in logic and proof writing.

Question 3. Write each of the following statements in the form 'if then':
1. You get a job implies that your credentials are good.
2. The banana trees will bloom, if stays warm for a month.
3. A quadrilateral is a parallelogram, if its diagonals bisect each other.
4. To get an A in the class, it is necessary that you do all exercises of the book.
Answer:
1. If you get a job, then your credentials are good.
2. If it stays warm for a month, then the banana tree will bloom.
3. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
4. If you want to get an A in the class, then you do all exercises of the book.
In simple words: We are changing each sentence to start with "If" and then say what "then" will happen. This means we are showing a cause and effect relationship for each statement.

Exam Tip: Always identify the cause (the condition) and the effect (the consequence) in the original statement to correctly structure the 'if-then' form. The part that implies or is necessary usually comes after 'if'.

Question 4. Given statements in (a) and (b), identify the statements given below as contrapositive or converse of each other:
(a) If you live in Delhi, then you have winter clothes.
(i) If you do not have winter clothes, then you do not live in Delhi.
(ii) If you have winter clothes, then you live in Delhi.
(b) If a quadrilateral is a parallelogram, then its diagonals bisect each other.
(i) If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a parallelogram.
(ii) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Answer:
(a)
(i) Contrapositive statement
(ii) Converse statement

(b)
(i) Contrapositive statement
(ii) Converse statement
In simple words: For part (a) and (b), we are checking if the new statements are either the 'contrapositive' (swapped and negative) or the 'converse' (just swapped) of the main statement.

Exam Tip: To identify a contrapositive, look for both the hypothesis and conclusion swapped, and both negated. For a converse, only the hypothesis and conclusion are swapped, with no negation. This distinction is vital for logical reasoning questions.