CBSE Class 11 Maths Revision Notes Chapter 14

Class 11 Mathematics  Revision Notes for Mathematical Reasoning of Chapter 14

Class 11 Mathematics Chapter 14 Notes of Mathematical Reasoning are designed for students looking to gain conceptual clarity on the topic. These notes are carefully prepared by subject-matter experts to make mathematical learning easy for students to prepare for their exams. 

Class 11 Mathematics  Revision Notes for Mathematical Reasoning of Chapter 14 – Free Download 

Access Class 11 Mathematics  Chapter 14 – Mathematical Reasoning Notes 

Basics:

Inductive and deductive reasoning are the two types of reasoning in Mathematics. 

Logic:

The way of reasoning is concerned with the study of logic. It gives guidelines for judging the appropriateness of a particular argument in the context of the theorem proof.

Statement (Proposition):

• The fundamental unit of mathematical reasoning is a mathematical statement.
• A sentence is said to be a mathematically acceptable statement if it is either true or false and not both.
• A statement is said to be an aggressive sentence if it is either true or false, but not both true and false. It is also known as a valid statement. Otherwise, it is known as an invalid statement.
• Statements are indicated in small letters. For example, p, q, z, etc.
• Example:

1. A dog has four legs.
2. Chemistry is an experimental and fun subject.
3. Mathematics is an interesting subject.

• An ambiguous sentence is not admissible as a statement in mathematics.
• The statement must be “mathematically acceptable.”

Open and Compound Statement 

• An open statement is a sentence with one or more variables that becomes a statement when certain values are assigned to them.
• A compound statement is created when two or more basic statements are joined together with terms like “and,” “if and only if,” “or,” “not,” “if,” or “then.”
• For example: 

1. My neighbour has a pet which is either a dog or a cat. 
2. The grass is green and the lake is blue.

The component statements here are- p: the lake is blue, q: the grass is green and the connecting word is ‘and’.  

• A compound statement containing ‘and’ is true only if all its component statements are true. 
• A component statement having ‘and’ is false if any of its component statements are false. This is so in both of these cases – when some of its component statements are false or all of its component statements are false.
• A compound statement with an ‘Or’ is true either when one of its component statements is true or both of its component statements are true. 
• A compound statement with an “or” is false if both of its component statements are wrong.
• There are two types of ‘Or’ statements: 

1. Exclusive “Or” – Example: Two lines intersect at a point or are parallel.
2. Inclusive “Or” – Example: We require a passport or a voter registration car to enter a country.

Elementary Operation of Logic 

• Conjunction – 

The conjunction of p and q is a compound sentence made up of two basic sentences, p, and q, joined together by the conjunction “and.” p∧q stands for a conjunction sentence.

• Disjunction –

The disjunction of p and q is a compound sentence made up of two simple sentences, p, and q, joined together by the conjunction “or.” pVq stands for a disjunction sentence.

• Negation – 

A statement that is formed by changing the truth value of a given statement by ‘no’, or ‘not’ is referred to as the negation of the given statement. If p is a statement, the negation of it is represented by ¬p 

For Example, 

p: Lucknow is a city. 

The negation of this statement is: 

• It is not the case that Lucknow is a city. 
• It is false that Lucknow is a city. 
• Lucknow is not a city.

• Conditional Sentence (Implication) –

The conditional sentence of p and q is made up of two simple statements, p and q, joined by the words “if” and “then.”  It is represented by the p⇒q sign. 

• Biconditional Sentence (Bi-implication) – 

Two simple sentences connected by the words, ‘if and only if’ are referred to as the biconditional sentence and are denoted by ‘⇔⇔’.

Tautology and Contradiction:

A compound statement that is true for every value of its components is known as a tautology.

The compound statements that are false for every value of their components are known as contradictions. They are also called a fallacy.

Quantifiers and Quantified Statements:

An open sentence is converted into a quantified statement by the use of a quantifier. In such remarks, two crucial symbols are employed:

• The universal quantifier is represented by the symbol “∀” which stands for “all values of.”
• The existential quantifier symbol, “∃” stands for “there exists.”

Example 

p: p is an irrational number for every prime number p. It means that if S denotes the set of all prime numbers, then all the members i.e. p of the set S are irrational numbers. 

Contrapositive and converse:

• Contrapositive and Converse are two specific alternative statements that can be created from a particular “if-then” expression.
• “If and only if,” denoted by the symbol “,” refers to the equivalent versions of the supplied statement and p as follows:

P is a necessary and sufficient condition for q and vice versa. 

P if and only if q 

q if and only if p

p⇔q

• Example 

1.  When the physical environment changes, the biological environment changes as well. 
2.  The converse of this statement is: If the biological environment does not shift, then neither will the physical environment.

Class 11 Mathematics Notes of Mathematical Reasoning 

It requires one to thoroughly have conceptual clarity and remember formulae to solve a given problem effortlessly in Mathematics. For this, students can use the revision notes to grasp concepts quickly and effectively. Class 11 Chapter 14 Mathematics Notes provided by Extramarks are curated by subject matter experts according to the updated CBSE syllabus. 

The Chapter 14 Mathematics Class 11 Notes of Mathematical Reasoning are available on the Extramarks’ website. Students can access it easily on their devices and refer to them according to their convenience. Having Mathematical Reasoning Class 11 Notes makes it simple for quick revision before examinations by referring to it. 

Class 11 Revision Notes Mathematical Reasoning 

Logic – It deals with the study of formulas for mathematical reasoning, that is based on statements or propositions. Mathematical reasoning is broadly divided into two types: 

• Inductive Reasoning – It is a non-rigorous method of reasoning based on generalised statements. In this, we check the validity of the statement against a set of rules. An example of inductive reasoning is given below:

1. A man’s neighbour’s cat hisses at him daily.
2. All the cats hiss at that man at the pet store. 

Hence, it can be deduced from the above statements that all cats hate that man.

• Deductive reasoning is a strict kind of reasoning in which claims are only accepted as true if the premise upon which they are based are true. Here, we begin with a premise or a few broad concepts, and then apply them to a particular circumstance. An example of deductive reasoning is below:

1. All humans have muscles.
2. All muscles are made up of living tissues.

With the help of the above two statements, it can be concluded that all humans are made up of living tissues.

Introduction to Important Topics in Class 11 Revision Notes Mathematical Reasoning 

1. Statements

A statement is defined as a clause that can be either true or false, but not simultaneously.

Points to Remember: No sentence can be a statement if:- 

• The sentence is explanatory.
• The sentence is in the form of a question.
• The sentence is an order or a request.

2. Simple Statements

If a sentence cannot be split down into two or more statements, it is said to be a simple statement.

3. Compound Statements

If a sentence is made up of two or more simple statements, it is said to be a compound statement. 

4. Connectives

Connectives are the terms that change or connect simple statements to form compound statements or new statements.

5. Conjunction 

The compound statement “a and b” that results when the words “and” are used to join two simple statements “a” and “b” is known as a conjunction of a and b. It is mathematically expressed as “a ∧ b” in symbolic form. 

Points to Remember: 

• When both a and b have the truth value T, the statement a ∧ b has the truth value T (true). 
• In the case of either a or b or both having the truth value F, the statement a ∧ b has the truth value F (false). 

6. Disjunction 

The compound statement “a or b,” which is created when two simple statements a and b are combined by the word “or,” is known as the disjunction of a and b. It is mathematically represented as “a ∨ b” in symbolic form.

Points to Remember:

• The statement a ∨ b holds the truth value F when both ‘a’ and ‘b’ have the truth value F. 
• The statement a ∨ b holds the truth value T when either ‘a’ or ‘b’ or both have the truth value T. 

7. Negation 

The negation of a statement refers to a postulation that the assertion is false or untrue. The sign denoting the negation of the statement “a” is “~a”.

Points to Remember: 

• When ‘a’ holds the truth value F, only then ~a holds the truth value T. 
• When ‘a’ holds the truth value T, only then ~a hold the truth value F. 

8. Negation of Conjunction 

The negation of a conjunction a ∧ b is said to be the disjunction of the negation of ‘b’ and the negation of ‘a’. Mathematically, we denote it as ~ (a ∧ b) = ~a ∨ ~b. 

9. Negation of Disjunction 

The negation of a disjunction a v b is said to be the conjunction of the negation of ‘b’ and the negation of ‘a’. Mathematically, we denote it as ~(a ∨ b) = ~a ∧ ~b. 

10. Negation of Negation 

The statement itself is what is termed ‘Negation of negation’ of a statement. Correspondingly, we denote it as ~(~a) = a 

11. The Conditional Statement 

If a and b are any two statements, then the compound statement, “if a then b”, formed by joining b and a by a connective ‘if-then’ is referred to as a conditional or an implication statement. It is mathematically denoted in the symbolic form of “a → b” or “a ⇒ b”. Here, ‘a’ is called the hypothesis or the antecedent, and ‘b’ is called the conclusion or the consequent of the conditional statement (a ⇒ b). 

12. Contrapositive of Conditional Statement 

The statement “(~b) → (~a)” is referred to as the contrapositive of the statement “a → b”. 

13. Converse of a Conditional Statement 

The conditional statement “a → b” is referred to as the converse of the conditional statement “a → b”.