Mathematical Reasoning - Notes, Overview, Books, FAQs

In Mathematics, Mathematical Reasoning is one of the easy topics to understand, Question is easy from this topic to solve. Every year you will get at max 1 - 2 questions in JEE Main and other exams, directly (as chapter weight in jee main is only 3%) but indirectly, the concept of this chapter will be involved in the physics semiconductor chapter where you will learn about gates and then the use of the truth table will be handy for you which you will learn in this chapter. The easy to score fact makes this chapter important from exam point of view as you will be able to solve the question in less than a minute if you have learned the concept and will be able to score some easy marks. It will be a new chapter for the student but you will find it quite easy to learn and understand. Once you through with concepts for this concept you will be able to solve questions mostly in mind with a little bit of calculation required on paper. Overall this chapter will be short compared to other chapters.

Why Mathematical Reasoning:

Mathematical Reasoning helps students to develop thinking ability and help them to learn how to think mathematically and approach a problem. It teaches you how to make sense of a given thing. These learning of sense-making can be used in the mathematical problem or can be applied in unfamiliar situations, These learning will be very helpful for future learning, to deduce the thing from top to bottom or induce some result from bottom to top.

If you understand this chapter well you will realize how good mathematics is and you will try to stay away from rote learning and that will help you to continue your journey with mathematics smoothly otherwise you may find mathematics frustrating in future going ahead.

After studying Mathematical Reasoning:

1. After studying this chapter you will be comfortable with logical reasoning, deductive reasoning, and sense-making thinking.

2. You will be able to solve questions related to logical thinking and deductive, inductive reasoning, you will feel confident about your way of thinking.

3. Since mathematical reasoning involves tautology as well as a truth table, so it will be helpful to you to understand gates in physics in semiconductor chapter, If you have studied this chapter before going to those chapter of physics.

4. And obviously, the chapter itself will help you to score some easy marks in the exam as it gets about 3% weight in jee main and you will be able to solve those questions very quickly if you have studied chapter well.

Notes on Mathematica Reasoning:

Important topics :

1. Statements and type of statements

2. Basic logical connectives, conjunction, and disjunction

3. Negation

4. Conditional statements

5. The contrapositive of conditional statements

6. The converse of conditional statements

7. Biconditional statements

8. Quantifiers

9. Validity of statements

Overview of the Mathematical Reasoning:

Statement: A statement is a sentence which is either true or false, but not both simultaneously.

Note: No sentence can be called a statement if

• (i) It is an exclamation

• (ii) It is an order or request

• (iii) It is a question

• (iv) It involves variable time such as ‘today’ , ‘tomorrow’ , ‘yesterday’ etc.

• (v) It involves variable places such as ‘here’ , ‘there’ , ‘everywhere’ etc.

• (vi) It involves pronouns such as ‘she’ , ‘he’ , ‘they’ etc.

Simple Statements: Simple statements are those which can’t be broken down into two or more sentences.

Basic logical connectives: There are many ways of combining simple statements to form new statements. The words which combine or change simple statements to form new statements or compound statements are called Connectives.

Negation: An assertion that a statement fails or denial of a statement is called the negation of the statement. The negation of a statement is generally formed by introducing the word “not” at some proper place in the statement or by prefixing the statement with “It is not the case that” or It is false that”.

Conditional Statements: if p and q are any two statements, then the compound statement “if p then q” formed by joining p and q by a connective if-then’ is called a conditional statement or an implication and is written in symbolic form as p → q or p ⇒ q. Here, p is called the hypothesis (or antecedent) and q is called a conclusion (or consequent) of the conditional statement (p ⇒ q):

The contrapositive of a conditional statement: The statement “(~ q) → (~ p)” is

called the contrapositive of the statement p → q

The converse of a conditional statement: The conditional statement “q → p” is called the converse of the conditional statement “ p → q ”

The biconditional statement: If two statements p and q are connected by the connective ‘if and only if’ then the resulting compound statement “p if and only if q” is called a biconditional of p and q and is written in symbolic form as p ↔ q.

Quantifiers: Quantifiers are the phrases like ‘These exist’ and “for every”.

For example, There exists a triangle whose all sides are equal.

The validity of statements: Validity of a statement means checking when the statement is true and when it is not true. This depends upon which of the connectives, quantifiers, and implication is being used in the statement.

For Example, Validity of statement with ‘AND’
To show statement r : p ∧ q is true, show statement ‘p’ is true and the statement ‘q’ is true.

How to prepare Mathematical Reasoning:

Mathematical Reasoning is a basic topic or you can say building block for logical-mathematical thinking development and is used in many other chapters also like semiconductors in physics, so you must be through with this chapter.

1. To understand this topic you have to go in the order of the name of the topic of this chapter means first you must study the statement and its type then negation and conditional statement
2. Start with understanding basic concepts like simple definitions (what is a statement, what is conditional statement), under each topic independently before moving to another by going through every concept.
3. Once you’re clear with basic concepts move to complex concepts, the contrapositive of a statement, the converse of the statement. In the end, you must be able to integrate all the topics to solve on the question.
4. After studying these concepts go through solved examples and then go to MCQ and practice the problem to make sure you understood the topic.
5. Solve the questions of the books which you are following and then go to previous year papers.
6. While going through concept make sure you understand the derivation of formulas and try to derive them by your own, as many times you will not need the exact formula but some steps of derivation will be very helpful to solve the problem if you understand the derivation it will boost your speed in problem-solving.
7. At the end of chapter try to make your own short notes for quick revision, make a list of formula to revise quickly before exams or anytime when you required to revise the chapter, it will save lots of time for you.

NCERT Notes Subject wise link:

• NCERT Notes for class 12 Physics.
• NCERT Notes for class 12 Chemistry.
• NCERT Notes for class 12 Mathematics.
• NCERT Notes for class 12 Biology.

Best books for the preparation of Mathematica Reasoning:

First, finish all the concept, example and questions given in NCERT Maths Book. You must be thorough with the theory of NCERT. Then you can refer to the book Algebra Arihant by Dr. SK goyal or RD Sharma or Cengage Mathematics Algebra but make sure you follow any one of these not all. Sets, Relations, and Functions are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require. In our view, the NCERT book will be enough for this chapter.

NCERT Solutions Subject wise link:

• NCERT solutions for class 12 Physics.
• NCERT solutions for class 12 Chemistry.
• NCERT solutions for class 12 Mathematics.
• NCERT solutions for class 12 Biology.
  
  

NCERT Exemplar Solutions Subject wise link:

• NCERT exemplar solutions for class 12 Physics.
• NCERT exemplar solutions for class 12 Chemistry.
• NCERT exemplar solutions for class 12 Mathematics.
• NCERT exemplar solutions for class 12 Biology.