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Edited By Ravindra Pindel | Updated on Jul 13, 2022 06:35 PM IST

We humans are superior to other species because of our ability to reason. Here, in the NCERT solutions for Class 11 Maths chapter 12 exercise 14.4 you will learn the process of reasoning especially in the context of mathematics. There are two kinds of reasoning:–

**Inductive****Deductive**

In mathematical induction, you have already learned about inductive reasoning in the context of mathematics. In exercise 14.4 Class 11 Maths you will learn about some fundamentals of deductive reasoning. In the previous exercises of this NCERT book chapter, you have already learned about the statements, negation of a statement, compound statements, special words and phrases, etc. In the Class 11 Maths chapter 14 exercise 14.4, you will learn about the implications of statements, contrapositive, and converse, etc. It is a very brief exercise but a very important exercise for mathematical reasoning. If you are looking for NCERT solutions for other classes you can click on the NCERT solutions link.

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**Also, see**

- Mathematical Reasoning Exercise 14.1
- Mathematical Reasoning Exercise 14.2
- Mathematical Reasoning Exercise 14.3
- Mathematical Reasoning Exercise 14.5
- Mathematical Reasoning Miscellaneous Exercise

Answer:

a.) If the square of a natural number is odd, then the natural number is odd.

b.) A natural number is not odd only if its square is not odd.

c.) For a natural number to be odd it is necessary that its square is odd.

d.) For the square of a natural number to be odd, it is sufficient that the number is odd

e.) If the square of a natural number is not odd, then the natural number is not odd.

Question:2.(i) Write the contrapositive and converse of the following statement.

If x is a prime number, then *x* is odd.

Answer:

The contrapositive is :

If a number x is not odd, then x is not a prime number.

The converse is :

If a number x in odd, then it is a prime number.

Question:2.(ii) Write the contrapositive and converse of the following statement.

If the two lines are parallel, then they do not intersect in the same plane.

Answer:

The contrapositive is:

If two lines intersect in the same plane, then they are not parallel.

The converse is:

If two lines do not intersect in the same plane, then they are parallel.

Question:2.(iii) Write the contrapositive and converse of the following statement.

Something is cold implies that it has low temperature.

Answer:

The contrapositive is:

If something is not at low temperature, then it is not cold.

The converse is:

If something is at low temperature, then it is cold .

Question:2.(iv) Write the contrapositive and converse of the following statement.

You cannot comprehend geometry if you do not know how to reason deductively.

Answer:

The contrapositive is:

If you know how to reason deductively, then you can comprehend geometry.

The converse is:

If you do not know how to reason deductively, then you cannot comprehend geometry.

Question:2.(v) Write the contrapositive and converse of the following statement.

x is an even number implies that x is divisible by 4.

Answer:

First, we convert the given sentence into the "if-then" statement:

If x is an even number, then x is divisible by 4.

The contrapositive is:

If x is not divisible by 4, then x is not an even number.

The converse is:

If x is divisible by 4, then x is an even number.

Question:3.(i) Write the following statement in the form “if-then”

You get a job implies that your credentials are good.

Answer:

The given statement in the form “if-then” is :

If you get a job, then your credentials are good.

Question:3.(ii) Write the following statement in the form “if-then”

The Banana tree will bloom if it stays warm for a month.

Answer:

The given statement in the form “if-then” is :

If the Banana tree stays warm for a month, then it will bloom.

Question:3.(iii) Write the following statement in the form “if-then”

A quadrilateral is a parallelogram if its diagonals bisect each other.

Answer:

The given statement in the form “if-then” is :

If diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Question:3.(iv) Write the following statement in the form “if-then”

To get an A + in the class, it is necessary that you do all the exercises of the book.

Answer:

The given statement in the form “if-then” is :

(iv) If you get A+ in the class, then you have done all the exercises in the book.

Question:4.(a) Given statements in (a). Identify the statements given below as contrapositive or converse of each other.

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.

Answer:

If __you live in Delhi__, then __you have winter clothes__. : (if p then q)

The Contrapositive is (~q, then ~p)

Hence (i) is the Contrapositive statement.

The Converse is (q, then p)

Hence (ii) is the Converse statement.

Question:4.(b) Given statements in (b). Identify the statements given below as contrapositive or converse of each other.

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:

If __a quadrilateral is a parallelogram__, then __its diagonals bisect each other__. (if p then q)

The Contrapositive is (~q, then ~p)

Hence (i) is the Contrapositive statement.

The Converse is (q, then p)

Hence (ii) is the Converse statement.

Class 11 Maths chapter 14 exercise 14.4 consists of questions related to implications of “if-then”, “only if” and “if and only if ” in the mathematical statements. In the Class 11 Maths chapter 14 exercise 14.4 you will also find questions related to contrapositive and converse in which statements can be formed from a given statement with “if-then”.

**Also Read| **Mathematical Reasoning Class 11 Notes

- Class 11 Maths chapter 14 exercise 14.4 solutions are designed by subject matter experts based on the guidelines given by CBSE.
- You can use these Class 11th Maths chapter 14 exercise 14.4 solutions for quick revision before the CBSE exam.

**Also see-**

- NCERT Solutions for Class 11 Maths
- NCERT Solutions for Class 11 Physics
- NCERT Solutions for Class 11 Chemistry
- NCERT Solutions for Class 11 Biology

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1. What is an statement ?

The mathematically acceptable statement is a sentence that is either true or false but not both.

2. Does the sentence "In 2003, the president of India was a woman" is an statement ?

The sentence "In 2003, the president of India was a woman" is a false sentence which means it is a statement.

3. Does the sentence "An elephant weighs more than a human being" is an statement ?

The sentence "An elephant weighs more than a human being" is a true sentence which means it is a statement.

4. Does the sentence "Women are more intelligent than men" is an statement ?

The sentence "Women are more intelligent than men" is not always true or false. This sentence is ambiguous which is not acceptable as a statement in mathematics.

5. Does the sentence "Two plus two equals four" is an statement ?

The sentence "Two plus two equals four" is always true which means there is no ambiguity regarding this sentence. Hence it is a statement in mathematics.

6. Does the sentence "All prime numbers are odd numbers" is an statement ?

The sentence "All prime numbers are odd numbers" is always false which means there is no ambiguity regarding this sentence. Hence it is a statement in mathematics.

Sep 06, 2024

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