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CBSE Class 11 Maths Notes Chapter 14 Mathematical Reasoning

Here, we have provided CBSE Class 11 Maths Notes Chapter 14 Mathematical Reasoning. Students can view these CBSE Class 11 Maths Notes Chapter 14 before exams for better understanding of the chapter.
authorImageAnanya Gupta9 May, 2024
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CBSE Class 11 Maths Notes Chapter 14

CBSE Class 11 Maths Notes Chapter 14: CBSE Class 11 Maths Notes Chapter 14 delves into the realm of Mathematical Reasoning, an essential aspect of problem-solving and logical thinking in mathematics.

This chapter aims to develop students ability to think logically and critically while analyzing mathematical statements and arguments. Mathematical reasoning involves identifying patterns, making conjectures, and formulating logical arguments to prove or disprove mathematical statements. By mastering mathematical reasoning, you'll improve your problem-solving skills and understand math concepts better.

CBSE Class 11 Maths Notes Chapter 14 PDF

You can find the CBSE Class 11 Maths Notes Chapter 14 on Mathematical Reasoning in PDF format through the link provided below. This chapter helps you understand the basics of logical thinking in math. It covers topics like statements, logical operations, methods of proof, and how to apply mathematical reasoning. By using this PDF, you can learn how to analyze math statements, recognize patterns, and make logical conclusions, which will improve your problem-solving skills in math.

CBSE Class 11 Maths Notes Chapter 14 PDF

CBSE Class 11 Maths Notes Chapter 14 Mathematical Reasoning

In CBSE Class 11 Maths Notes Chapter 14 on Mathematical Reasoning, you'll find solutions to various problems related to logical thinking and mathematical reasoning. This chapter helps you understand how to approach and solve problems using logical deductions, inference, and reasoning skills. It covers topics like propositional logic, mathematical statements, logical connectives, truth tables, and methods of proof. By studying these solutions, you'll develop a deeper understanding of logical reasoning and its applications in mathematics, which will enhance your problem-solving abilities.

Basics

Inductive and deductive reasoning are two fundamental types of reasoning in mathematics.

Inductive reasoning involves making generalizations based on specific observations or patterns. It starts with specific examples and then generalizes to broader conclusions. For example, if you observe that a series of numbers follows a certain pattern, you might use inductive reasoning to generalize that pattern to predict the next numbers in the series.

On the other hand, deductive reasoning involves deriving specific conclusions from general principles or premises. It starts with general principles or assumptions and then applies logical rules to reach specific conclusions. For example, if you know that all squares have four sides and you're given that a shape is a square, deductive reasoning allows you to conclude that it must have four sides. Both types of reasoning are essential in mathematics and are used in various problem-solving scenarios to derive new results and make logical arguments.

Logic

The study of logic is indeed focused on the principles and rules that govern reasoning and argumentation. Logic provides us with a systematic framework for evaluating the validity of arguments and determining whether conclusions logically follow from premises. It helps us understand how to construct sound arguments and avoid common pitfalls in reasoning. In the context of theorem proof, logic plays a crucial role in ensuring the validity and rigor of mathematical arguments. By applying logical principles, mathematicians can construct and analyze proofs to establish the truth or validity of mathematical statements. In essence, logic serves as a foundational tool for mathematicians, providing them with the means to reason effectively and rigorously in their pursuit of mathematical truth.

Statement (Proposition)

Mathematical statements serve as the building blocks of mathematical reasoning, providing the basis for logical arguments and deductions. A mathematically acceptable statement must possess the property of being definitively true or false, without any ambiguity or uncertainty. This characteristic distinguishes it from non-mathematical statements, which may be subjective or open to interpretation. In mathematical discourse, statements are typically represented by lowercase letters such as p, q, r, etc., allowing for concise and systematic notation.

For Example:

  • "A cow has four legs" is a mathematically acceptable statement since it can be definitively verified as true or false.
  • "Chemistry is an experimental subject" is also a valid statement, as its truth or falsehood can be determined based on empirical evidence.
  • "Maths is a fun subject" may be a subjective opinion rather than a factual statement and therefore does not qualify as a mathematically acceptable statement.
Ambiguity or vagueness in a statement renders it invalid for mathematical purposes, highlighting the importance of precision and clarity in mathematical discourse. Only statements that meet the criteria of being mathematically acceptable can serve as reliable foundations for logical reasoning and deduction.

Open and Compound Statement

An open statement is one that contains variables, requiring specific values to be assigned to them in order to become a definitive statement. When simple statements are combined using words like 'and', 'or', 'not', 'if', 'then', or 'if and only if', they form compound statements. For instance:
  • "That house has a pet which is either a dog or a cat" is a compound statement composed of two simple statements connected by 'or'.
  • "The lake is blue and the grass is green" combines two simple statements with 'and'.
The truth value of a compound statement depends on the truth values of its component statements. For compound statements with 'and', all component statements must be true for the compound statement to be true. Conversely, for compound statements with 'or', the compound statement is true if at least one of its component statements is true. Inclusive 'or' allows for the possibility that both component statements may be true, while exclusive 'or' indicates that only one of the component statements can be true, but not both. For example:
  • Inclusive 'or': "To enter a country, we require a passport or a voter registration card."
  • Exclusive 'or': "Two lines intersect at a point or are parallel."

Elementary Operation of Logic

Conjunction: When two simple sentences, p and q, are combined using 'and', the resulting compound sentence is called a conjunction, represented by p∧q.

Disjunction: If two simple sentences, p and q, are connected using 'or', the resulting compound sentence is termed a disjunction, represented by p∨q.

Negation: A statement formed by changing the truth value of a given statement, usually by using words like 'no' or 'not', is known as the negation of the given statement. If p is a statement, its negation is denoted by ¬p.

Example:

Consider the statement "Thane is a city." Its negation can be expressed as "It is not the case that Thane is a city," "It is false that Thane is a city," or simply "Thane is not a city."

Conditional Sentence (Implication): When two simple sentences, p and q, are linked by the phrase "if and then," it forms a conditional sentence denoted by p⇒q.

Biconditional Sentence (Bi-implication): I f two simple sentences are connected by the phrase "if and only if," it results in a biconditional sentence represented by the symbol '⇔'.

Table for Basic Logical Connections

Truth Value and Truth Table

A statement can be classified as either "true" or "false," represented by the symbols T and F, respectively, known as truth values. A truth table is a systematic list showing all possible truth values for the resulting statements, usually for different combinations of variables. In a compound statement, values are assigned to the variables involved, and the resulting truth value is determined accordingly. The number of rows in a truth table corresponds to the number of possible combinations of truth values for the variables.

Tautology and Contradiction

A compound statement is termed a tautology if it is true for every possible combination of truth values of its components. A compound statement is termed a contradiction (or fallacy) if it is false for every possible combination of truth values of its components.

Truth Table:

Quantifiers and Quantified Statements

In mathematical logic, two important symbols are employed:
  1. The symbol ‘∀’ represents ‘for all’ or ‘all values of’ and is known as the universal quantifier.
  2. The symbol ‘∃’ represents ‘there exists’ and is called the existential quantifier.
For instance: p: For every prime number p, p is an irrational number. This statement means that if S denotes the set of all prime numbers, then for all members p of the set S, p is an irrational number.

Quantified Statement

A quantifier converts an open sentence into a statement, forming what is known as a quantified statement.

Negation of a Quantified Statement

The negation of a universally quantified statement is an existential quantified statement, and vice versa.

Implications

Statements with expressions like “if-then,” “only if,” and “if and only if” are termed implications.

Example: r: If a number is a multiple of 9, then it is a multiple of 3. This statement can be represented as p ⇒ q, meaning p implies q. Here, p is a sufficient condition for q. Alternatively, q is a necessary condition for p.

Contrapositive and Converse

Contrapositive and converse are alternative statements derived from an original statement with “if-then” logic. ‘If and only if’, denoted by the symbol ‘⇔’, implies equivalence between two statements p and q.

Example:

Original Statement

When the physical environment shifts, the biological environment shifts as well.

Converse

If the biological environment does not change, then neither does the physical environment.

Validating Statements

Various rules guide the assessment of statement validity.

Rule 1: For statements with “And”, both component statements must be true.

Rule 2: For statements with “Or”, either component statement must be true.

Rule 3: For statements with “If-then”, either a direct or contrapositive method can be used to establish validity.

Rule 4: Statements with “if and only if” require validation in both directions.

By Contradiction

To confirm the truth of a statement p, we assume its negation, ¬p, is true. If this assumption leads to a contradiction, we conclude that p is true. This method may involve providing a counterexample where the statement is false.

Benefits of CBSE Class 11 Maths Notes Chapter 14 Mathematical Reasoning

Conceptual Clarity: The notes provide clear explanations of mathematical reasoning concepts, helping students understand the fundamental principles underlying logical thinking in mathematics.

Structured Learning: The notes are organized systematically, covering various topics such as statements, quantifiers, implications, and valid reasoning methods. This structured approach aids in better comprehension and retention of the subject matter.

Problem-Solving Skills: Through examples and exercises, students learn how to apply mathematical reasoning to solve problems effectively. This enhances their analytical and problem-solving skills, which are valuable not only in mathematics but also in other areas of study and real-life situations.

Preparation for Exams: The notes cover topics aligned with the CBSE Class 11 Maths curriculum, helping students prepare comprehensively for their exams. By studying these notes, students can strengthen their understanding of mathematical reasoning concepts and perform better in assessments.

Critical Thinking Development: Mathematical reasoning encourages critical thinking and logical reasoning abilities. By engaging with the concepts presented in the notes, students develop skills such as analyzing information, evaluating arguments, and making informed decisions, which are essential in various academic and professional pursuits.

CBSE Class 11 Notes for Maths
Chapter 1 Sets Notes Chapter 2 Relations and Functions Notes
Chapter 3 Trigonometric Functions Notes Chapter 4 Principle of Mathematical Induction Notes
Chapter 5 Complex Numbers and Quadratic Equations Notes Chapter 6 Linear Inequalities Notes
Chapter 7 Permutations and Combinations Notes Chapter 8 Binomial Theorem Notes
Chapter 9 Sequences and Series Notes Chapter 10 Straight Lines Notes
Chapter 11 Conic Sections Notes Chapter 12 Introduction to Three Dimensional Geometry Notes
Chapter 13 Limits and Derivatives Notes Chapter 14 Mathematical Reasoning Notes
Chapter 15 Statistics Notes Chapter 16 Probability Notes

CBSE Class 11 Maths Notes Chapter 14 FAQs

What is mathematical reasoning?

Mathematical reasoning involves using logical arguments and principles to analyze and solve mathematical problems. It includes understanding mathematical statements, quantifiers, implications, and valid reasoning methods.

Why is mathematical reasoning important?

Mathematical reasoning is essential for developing critical thinking skills, problem-solving abilities, and logical reasoning capabilities. It helps students understand the underlying principles of mathematics and apply them to various real-life situations.

What are quantifiers in mathematical reasoning?

Quantifiers, such as "for all" (∀) and "there exists" (∃), are used to specify the scope of mathematical statements. They indicate whether a statement applies to all elements in a set (universal quantifier) or if there exists at least one element in a set satisfying a certain property (existential quantifier).

How can I prepare for exams based on Chapter 14 of CBSE Class 11 Maths?

To prepare for exams, students should thoroughly study the concepts covered in Chapter 14, practice solving problems and exercises from their textbook and class notes, and review sample question papers and past exam papers to familiarize themselves with the exam pattern and question types.
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