CBSE Class 11 Mathematics Chapter 14 Mathematical Reasoning NCERT Solutions
This chapter on Mathematical Reasoning for CBSE Class 11 Mathematics introduces students to the fundamental concepts of logic. The NCERT Solutions provided here cover Exercise 14.1, focusing on identifying statements and understanding why certain sentences are not statements. Students will learn to distinguish between declarative sentences that are either true or false and those that are commands, questions, opinions, or ambiguous. The solutions explain the criteria for a sentence to be a statement, emphasizing its truth value. This foundational knowledge is crucial for understanding more complex mathematical arguments and proofs in higher studies. These solutions are designed to help students grasp these concepts clearly, aiding in their exam preparation and building a strong base in logical thinking.
Quick info
| Board | CBSE |
|---|---|
| Class | Class 11 |
| Subject | Mathematics |
| Session | 2026 |
| Language | English |
| Type | NCERT Solutions |
| Chapter | Chapter 14 |
Chapter summary
Chapter 14, Mathematical Reasoning, focuses on the basic principles of logical statements. The NCERT Solutions for this chapter help students differentiate between statements (which have a definite truth value) and non-statements (like questions, commands, opinions, or ambiguous sentences). It covers the criteria for a sentence to be considered a statement and provides examples to illustrate these concepts, preparing students for logical analysis in mathematics.
Learning outcomes
- Understand the definition of a mathematical statement.
- Identify sentences that are statements and provide reasons.
- Distinguish between statements and non-statements (questions, commands, opinions).
- Analyze the truth value of given sentences.
- Apply the criteria for a statement to new examples.
Topics covered
Paper topics
- Mathematical Statements
- Truth Value
- Declarative Sentences
- Identifying Statements
- Non-Statements
- Questions as Non-Statements
- Commands as Non-Statements
- Opinions as Non-Statements
- Ambiguous Sentences
- Logical Reasoning Basics
Important topics
- Definition of a Statement
- Criteria for a Statement
- Distinguishing Statements from Non-Statements
- Truth Value Analysis
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Questions and Solutions
Question 1
(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.
To determine if a sentence is a statement, we must check if it is a declarative sentence that has a definite truth value (i.e., it is either true or false).
(i) There are 35 days in a month.
This sentence is a declarative statement. Although it is factually incorrect (months have at most 31 days), it has a definite truth value (False). Therefore, it is a statement.
(ii) Mathematics is difficult.
This sentence expresses an opinion. The difficulty of mathematics is subjective and varies from person to person. It cannot be definitively classified as true or false for everyone. Therefore, it is not a statement.
(iii) The sum of 5 and 7 is greater than 10.
This is a declarative sentence. The sum of 5 and 7 is 12, and 12 is indeed greater than 10. Since the sentence is true, it has a definite truth value (True). Therefore, it is a statement.
(iv) The square of a number is an even number.
This sentence is not a statement because its truth value is not definite. For example, the square of 2 is 4 (even), but the square of 3 is 9 (odd). Since it is sometimes true and sometimes false depending on the number chosen, it is not a statement.
(v) The sides of a quadrilateral have equal length.
This sentence is not a statement. While some quadrilaterals (like squares and rhombuses) have equal sides, others (like rectangles and trapeziums) do not. The truth of the statement depends on the specific type of quadrilateral, making its truth value indefinite in a general context. Therefore, it is not a statement.
(vi) Answer this question.
This sentence is an imperative command, not a declarative sentence. It asks someone to perform an action. Therefore, it is not a statement.
(vii) The product of (-1) and 8 is 8.
This is a declarative sentence. The product of (-1) and 8 is actually -8. Since the sentence states something that is factually incorrect, it has a definite truth value (False). Therefore, it is a statement.
(viii) The sum of all interior angles of a triangle is 180°.
This is a well-known geometric fact and a declarative sentence. It is always true for any triangle. Therefore, it is a statement.
(ix) Today is a windy day.
This sentence is not a statement because its truth value is ambiguous and depends on context. 'Today' and 'windy' are relative terms. Without a specific date and location, and a clear definition of 'windy', its truth cannot be determined. Therefore, it is not a statement.
(x) All real numbers are complex numbers.
This is a declarative sentence. Every real number 'a' can be expressed as a complex number 'a + 0i'. Thus, the statement is always true. Therefore, it is a statement.
Question 2
A sentence is not a statement if it does not have a definite truth value (i.e., it is not clearly true or false). This can happen if the sentence is a question, a command, an exclamation, an opinion, or is ambiguous.
Here are three examples of sentences that are not statements:
(i) He is a doctor.
Reason: This sentence is not a statement because it is ambiguous. The pronoun 'He' does not refer to a specific person, so we cannot determine whether the statement is true or false without more context.
(ii) Geometry is difficult.
Reason: This sentence is not a statement because it expresses a subjective opinion. What one person finds difficult, another might find easy. It does not have a definite truth value that applies universally.
(iii) Where is she going?
Reason: This sentence is not a statement because it is a question. Questions seek information and do not assert a proposition that can be true or false.
Common mistakes
- Confusing opinions or subjective statements with declarative statements.
- Failing to recognize that commands and questions are not statements.
- Not considering the definiteness of the truth value for a sentence to be a statement.
- Assuming a sentence is a statement if it appears factual without checking for ambiguity.
Revision tips
- Focus on the core definition: a statement must be either true or false, and not both.
- Practice identifying the reason why a sentence is NOT a statement (e.g., it's a question, an opinion, or ambiguous).
- Review the examples provided in the solutions to solidify your understanding.
- Try creating your own examples of statements and non-statements.
Practice MCQs
Q1. Which of the following is a statement?
Explanation: A statement must have a definite truth value. 'The Earth is flat' is a declarative sentence that is definitively false, making it a statement. The others are commands, questions, or opinions.
Q2. Why is the sentence 'Mathematics is difficult' not a statement?
Explanation: The difficulty of mathematics is subjective and varies from person to person, meaning it does not have a universally true or false value.
Q3. The sentence 'The sum of 5 and 7 is 12' is:
Explanation: This sentence is a statement because it is a declarative sentence that is unequivocally true.
Q4. Which of the following is NOT a statement?
Explanation: 'Answer this question' is a command, not a declarative sentence with a truth value, hence it is not a statement.
Q5. A sentence is considered a statement in logic if it is:
Explanation: The fundamental criterion for a sentence to be a statement is that it must possess a definite truth value, meaning it can be classified as either true or false.
Frequently asked questions
What is a statement in Mathematical Reasoning?
In Mathematical Reasoning, a statement is a declarative sentence that is either true or false, but not both. It must have a definite truth value.
Why are questions and commands not considered statements?
Questions and commands are not statements because they do not assert anything that can be definitively classified as true or false. They are requests for information or actions.
How can I determine if a sentence is a statement?
Check if the sentence is a declarative statement (it states a fact or assertion) and if it has a definite truth value (it is either true or false, without ambiguity).
What does it mean for a sentence to have a 'truth value'?
A truth value means that the sentence can be definitively labeled as either true (T) or false (F). If a sentence's truth cannot be determined or it can be both true and false, it does not have a truth value and is not a statement.
Are opinions statements in logic?
No, opinions are generally not statements in logic because they are subjective and do not have a universally agreed-upon truth value. For example, 'This movie is boring' is an opinion.
How do these NCERT solutions help with Chapter 14?
These solutions provide clear, step-by-step explanations for identifying statements and non-statements, helping students understand the core concepts of mathematical logic and prepare for exams.
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