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NCERT Class 11 Maths Textbook Chapter 14 With Answer Book PDF Free Download
Chapter 14: Mathematical Reasoning
14.1 Introduction
In this chapter, we shall discuss some basic ideas of Mathematical Reasoning. All of us know that human beings evolved from the lower species over many millennia.
The main asset that made humans “superior” to other species was the ability to reason.
How well this ability can be used depends on each person’s power of reasoning. How do develop this power? Here, we shall discuss the process of reasoning, especially in the context of mathematics.
In mathematical language, there are two kinds of reasoning – inductive and deductive. We have already discussed inductive reasoning in the context of mathematical induction.
In this chapter, we shall discuss some fundamentals of deductive reasoning.
14.2 Statements
The basic unit involved in mathematical reasoning is a mathematical statement. Let us start with two sentences:
In 2003, the president of India was a woman.
An elephant weighs more than a human being
When we read these sentences, we immediately decide that the first sentence is false and the second is correct. There is no confusion regarding these. In mathematics, such sentences are called statements. On the other hand, consider the sentence:
Women are more intelligent than men.
Some people may think it is true while others may disagree. Regarding this sentence, we cannot say whether it is always true or false. That means this sentence is ambiguous. Such a sentence is not acceptable as a statement in mathematics.
| Author | NCERT |
| Language | English |
| No. of Pages | 26 |
| PDF Size | 1738 KB |
| Category | Mathematics |
| Source/Credits | ncert.nic.in |
NCERT Solutions Class 11 Maths Chapter 14 Mathematical Reasoning
- Which of the following sentences are statements? Give reasons for your answer.
(i) There are 35 days in a month.
(ii) Mathematics is difficult.
(iii) The sum of 5 and 7 is greater than 10.
(iv) The square of a number is an even number.
(v) The sides of a quadrilateral have equal length.
(vi) Answer this question.
(vii) The product of (–1) and 8 is 8.
(viii) The sum of all interior angles of a triangle is 180°.
(ix) Today is a windy day.
(x) All real numbers are complex numbers.
Solution:
(i) The maximum number of days in a month is 31, so this sentence is incorrect. Therefore it is a statement
(ii) This sentence is subjective. For some people, Mathematics can be easy and for some others, it can be difficult. Therefore it is not a statement
(iii) The sum of 5 and 7 is 12 and it is greater than 10. Therefore this sentence is always correct. Hence, it is a statement
(iv) This sentence can be sometimes correct and sometimes incorrect. For example, the square of 2 is an even number but the square of 3 is an odd number. Hence, it is not a statement
(v) This sentence can be sometimes correct and sometimes incorrect. For example, square and rhombus have sides of equal lengths whereas trapezium and rectangle have sides of unequal lengths. Therefore it is not a statement
(vi) It is an order. Hence, it is not a statement
(vii) The given sentence is incorrect because the product of (-1) and 8 is – 8. Hence, it is a statement
(viii) The given sentence is correct and therefore, it is a statement
(ix) The given sentence is not a statement because the day that is being referred to is not evident from the sentence.
(x) The given sentence is always correct because all real numbers can be written as a × 1 + 0 × i. Hence, it is a statement.
2. Give three examples of sentences that are not statements. Give reasons for the answers.
Solution:
The three examples of sentences, which are not statements are given below:
(i) He is a doctor
In the given sentence, it is not evident as to whom ‘he’ is referred to. Hence, it is not a statement
(ii) Geometry is difficult
For some people, geometry can be easy and for some others, it can be difficult. Hence, this is not a statement
(iii) Where is she going?
In this question, it is not evident to whom ‘she’ is referred. Hence, it is not a statement
Exercise 14.2
1. Write the negation of the following statements:
(i) Chennai is the capital of Tamil Nadu.
(ii) is not a complex number.
(iii) All triangles are not an equilateral triangles.
(iv) The number 2 is greater than 7.
(v) Every natural number is an integer.
Solution:
(i) Chennai is not the capital of Tamil Nadu
(ii)
is a complex number
(iii) All triangles are equilateral triangles
(iv) The number 2 is not greater than 7
(v) Every natural number is not an integer
2. Are the following pairs of statements negations of each other?
(i) The number x is not a rational number.
The number x is not an irrational number.
(ii) The number x is a rational number.
The number x is an irrational number.
Solution:
(i) The negation of the first statement is ‘the number x is a rational number’.
This is the same as the second statement because if a number is not an irrational number then the number is a rational number
Hence, the given statements are negations of each other
(ii) The negation of the first statement is ‘the number x is not a rational number. This means that the number x is an irrational number which is the same as the second statement.
Hence, the given statements are negations of each other
3. Find the component statements of the following compound statements and check whether they are true or false.
(i) Number 3 is prime or it is odd.
(ii) All integers are positive or negative.
(iii) 100 is divisible by 3, 11, and 5.
Solution:
(i) The component statements are
(a) Number 3 is prime
(b) Number 3 is odd
Here, both the statements are true
(ii) The component statements are
(a) All integers are positive
(b) All integers are negative
Here, both statements are false.
(iii) The component statements are
(a) 100 is divisible by 3
(b) 100 is divisible by 11
(c) 100 is divisible by 5
Here, statements (a) and (b) are false and (c) is true.
NCERT Class 11 Maths Textbook Chapter 14 With Answer Book PDF Free Download