Mathematical Reasoning Class 11 Notes CBSE Maths Chapter 14 [Free PDF Download]

Section–A (1 Mark Questions)

1. Identify the quantifier in the following statement: “There exists a real number x such that $\sqrt{x}$  is a natural number”.

Ans. The given statement contains existential quantifier ‘there exists’.

2. What is the Negation of “2+3=5 and 8<10”

Ans. Let p:2+3=5, q:8<10

Given proposition is: $p\wedge q$ 

Its negation is $\sim (p\wedge q)=\sim p\vee \sim q$ 

$\therefore$ we have $2+3\neq 5$ or $8\nless 10$.

3. What is the contrapositive statement of “If a triangle is equilateral, then it is isosceles”.

Ans. p: triangle is equilateral, 

q: triangle is isosceles 

$\sim p$:triangle is not equilateral, $\sim q$ : triangle is not isosceles 

The contrapositive of the statement $p\Rightarrow q$ is $\sim q\Rightarrow \sim p$ 

It’s contrapositive statement is:

‘If a triangle is not isosceles, then it is not equilateral’.

4. Write the converse statement of “If a number is divisible by 9, then it is divisible by 3”.

Ans. p: number is divisible by 9  ,  q: number is divisible by 3

The converse of p $\Rightarrow$  q is q $\Rightarrow$p

It’s converse statement is: ‘If a number is divisible by 3, then it is divisible by 9’. 

5. The quantifier in ‘Every city in U.S.A. is clean’ is ........... and it’s type is.......

Ans. Every’ and it’s type is ‘Universal’. 

Section–B (2 Marks Questions)

6. Which of the following sentences are statements? Give reasons for your answer:

(i)   The sum of 5 and 7 is greater than 10.

(ii)  There are 35 days in a month.

(iii) Mathematics is difficult.

(iv) Mathematics is a fun.

Ans. (i) The sentence is true because 5+7=12

Which is greater than 10. 

Hence, it is a statement.

(ii)  The sentence is false because a month cannot have more than 31 days. 

Hence, it is a statement.

(iii) The sentence is not a statement. 

($\because$ for some mathematics is easy and for others it is difficult)

(iv) The sentence is subjective. Mathematics is a fun for those who like it but it is not fun who do not like it.

Hence, it is not a statement.

7. Write the negation of the following statement:Paris is in France and London is in England.

Ans. Let p: Paris is in France and 

q: London is in England.

$\because$ The conjunction is given by $p\wedge q$

Now $\sim p:$ Paris is not in France 

And $\sim q:$  London is not in England.

$\because$ Negation of $p\wedge q$ is given by:

 $\sim (p\wedge q):$ Paris is not in France or London is not in England.

8. Write the Negation of the proposition: If we control population growth, we prosper.

Ans. p: we control population, 

q: we prosper 

 $\because$ We have $p\Rightarrow q$

Its negation is $\sim (p\Rightarrow q)$ i.e., $p\wedge \sim q$ 

i.e. we control population but we do not prosper. 

9. Write the contrapositive of ‘if two triangles are identical, then these are similar’.

Ans. Consider the following statements

p: Two triangles are identical. 

q: Two triangles are similar.

Clearly, the given statement in symbolic form is $p\Rightarrow q$ . 

Therefore, its contrapositive is given by $\sim q\Rightarrow \sim p$ 

Now, $\sim p:$ two triangles are not identical.

 $\sim q:$ two triangles are not similar.

Therefore, $\sim q\Rightarrow \sim p:$ If two triangles are not similar, then these are not identical.

10. Identify the quantifier in each of the following statement:

(i)   For every real number x, x+3 > x

(ii) There exists a real number x such that $\sqrt{x}$  is natural number.

(iii) Some real numbers are rational.

(iv) All birds can fly.

Ans. (i)   The given statement contains universal quantifier ‘for every’.

(ii)  The given statement contains existential quantifier ‘there exists’.

(iii) The given statement contains existential quantifier ‘some’. 

(iv)  The given statement contains universal quantifier ‘all’.

11. Explain open statement with an example.

Ans. Open Statement: A declarative sentence involving variable(s) is said to be an open statement if the variable(s) is (are) replaced by a definite value(s).

For example:

 x is an integer.

This is true when x= ……-3, -2, -1, 0, 1, 2, 3….

but false when $x=\frac{2}{3},\frac{5}{7},-\frac{9}{5}....$.

12. Check the validity of the statement: “100 is a multiple of 4 and 5’’.

Ans. Let $p:r\wedge s$.

Where r : “100 is a multiple of 4” is true 

And s : “100 is a multiple of 5” is true 

Hence p is true. 

13. Translate the following statement into symbolic form:“Jack and Jill went up the hill”.

Ans. The given statement can be restated as:

“Jack went up the hill and Jill went up the hill”.

Let p : Jack went up the hill and q : Jill went up the hill.

 $\therefore$ The given statement, in symbolic form, is $p\wedge q$ .

PDF Summary - Class 11 Maths Mathematical Reasoning (Chapter 14)

Basics:

There are two types of reasoning in maths: inductive and deductive reasoning.

Logic:

• The study of logic is concerned with the way of reasoning.

• It gives us guidelines for judging the correctness of a particular argument in the context of theorem proof.

Statement (Proposition):

• A mathematical statement is the fundamental unit of mathematical reasoning.

• A sentence is called a mathematically acceptable statement if it is either true or false but not both.

• A statement is an aggressive sentence that is either true or false, but not both true and false which is known as valid statement. Otherwise, it is referred to as an invalid statement.

• Small letters are used to indicate statements.

For example, \[\text{p, q, r}\]etc.

• Example:

1. A cow has four legs.

2. Chemistry is an experimental subject.

3. Maths is a fun subject.

• In mathematics, an ambiguous sentence is not admissible as a statement.

• The statement should be "mathematically acceptable."

Open and Compound Statement

• An open statement is a sentence with one or more variables that becomes a statement when specified values are assigned to the variables.

• If two or more simple statements are combined by the use of words such as ‘and’, ‘or’, ‘not’, ‘if’, ‘then’, ‘if and only if ‘, then the resulting statement is called a compound statement.

• Example:

1. That house has a pet which is either a dog or a cat.

2. The lake is blue and the grass is green. 

3. The component statements are

• \[\text{p:}\]The lake is blue.

• \[\text{q:}\] The grass is green.

• The connecting word is ‘and’. 

• The compound statement which has  ‘And’ is true if all its component statements are true.

• The component statement wich has ‘And’ is false if any of its component statements is false (this includes the case that some of its component statements are false or all of its component statements are false).

• A compound statement with an ‘Or’ is true when one component statement is true or both the component statements are true.

• A compound statement with an ‘Or’ is false when both the component statements are false

• In Or statement there are two types

1. Exclusive “Or”

Example:

Two lines intersect at a point or are parallel

2.  Inclusive “Or”

 Example:

To enter a country, we require a passport or a voter registration card.

Elementary Operation of Logic

• Conjunction

• A compound sentence formed by two simple sentences \[\text{p}\]and \[\text{q}\] using connective ‘and’ is called the conjunction of \[\text{p}\] and \[\text{q}\] and it is represented by \[\text{p}\wedge \text{q}\].

• Disjunction

• A compound sentence formed by two simple sentences \[\text{p}\] and \[\text{q}\] using connectives ‘or’ is called the disjunction of \[\text{p}\] and \[\text{q}\] and it is represented by \[\text{p}\vee \text{q}\].

• Negation

• A statement which is formed by changing the truth value of a given statement by using the word like ‘no’, ‘not’ is called negation of given statement.

• If \[\text{p}\] is a statement, the negation of \[\text{p}\] is denoted by $\neg p $.

• Example

1. \[\text{p:}\]Thane is a city.

2. The negation of this statement is

    • It is not the case that Thane is a city

    • It is false that Thane is a city.

    • Thane is not a city.

• Conditional Sentence (Implication):

Two simple sentences \[\text{p}\] and \[\text{q}\] connected by the phase, if and then, is called conditional sentence of \[\text{p}\] and \[\text{q}\] and it is denoted by \[\text{p}\Rightarrow \text{q}\].

• Biconditional Sentence (Bi-implication):

The two simple sentences are connected by the phrase, ‘if and only if this is called biconditional sentences. 

It is denoted by the symbol ‘\[\Leftrightarrow \]’.

Table for Basic Logical Connections:

Truth Value and Truth Table:

• A statement can be either "true" or "false," which are represented by the symbols \[T\] and \[F\], respectively, and are termed truth values.

• A truth table is a list of all possible truth values for the resulting statements.

• Values are assigned to the variables in a compound statement.

• The number of rows is proportional to the number of statements.

Tautology and Contradiction:

• The compound statement which are true for every value of their components are called tautology.

• The compound statements which are false for every value of their components are called contradiction (fallacy).

Truth Table:

Quantifiers and Quantified Statements:

 In these statements, there are two important symbols are used.

• The symbol ‘\[\forall \]’ stands for ‘all values of ‘ and is called as universal quantifier.

• The symbol ‘\[\exists \]’ stands for ‘there exists’. The symbol \[\exists \] is known as existential quantifier.

Example

\[p:\] For every prime number \[p\], \[p\] is an irrational number.

It means that if \[S\] denotes the set of all prime numbers, then for all the

members’ \[p\] of the set \[S\], \[p\] is an irrational number.

Quantified Statement:

• A quantifier transforms an open sentence into a statement, which is referred to as a quantified statement.

Negation of a Quantified Statement:

• \[\sim\left\{ \forall \text{ x}\in \text{A}:\text{p}\left( \text{x} \right)\text{ is true} \right\}=\left\{ \exists \text{ x}\in \text{A such that }\left( \text{s}\text{.t} \right)\sim\text{p}\left( \text{x} \right)\text{ is true} \right\}\]

• \[\sim\left\{ \exists \text{ x}\in \text{A p}\left( \text{x} \right)\text{ is true} \right\}=\left\{ \forall \text{ x}\in \text{A : }\sim\text{p}\left( \text{x} \right)\text{ is true} \right\}\]

Implications:

• These are statements that have words “if-then”, “only if” and “if and only if”.

• Example

• \[r:\] If a number is a multiple of \[9\], then it is a multiple of  \[3\].

• Let \[p\] and \[q\] denote the statements

• \[p:\] a number is a multiple of \[9\].

• \[q:\] a number is a multiple of \[3\].

Then, if \[p\] then \[q\] is the same as the following:

• \[p\] implies \[q\] is denoted by \[p\Rightarrow q\]. The symbol \[\Rightarrow \] stands for implies.

• \[p\] is a sufficient condition for \[q\].

• \[p\] only if \[q\].

• \[q\] is a necessary condition for \[p\].

• $\neg q$ implies $\neg p$.

Contrapositive and converse:

• Contrapositive and converse are certain other statements that can be formed from a given statement with “if-then”.

• ‘If and only if’, represented by the symbol ‘\[\Leftrightarrow \]‘ means the following equivalent forms for the given statements \[p\] and q.

• \[p\] if and only if \[q\]

• \[q\] if and only if \[p\]

• \[p\] is necessary and sufficient condition for \[q\] and vice-versa

• \[p\Leftrightarrow q\]

• Example

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

• The converse of this statement is: If the biological environment does not change, then neither does the physical environment.

Validating Statements:

There are various general rules to check whether a statement is true or not.

• Rule \[1\]:

If \[p\] and \[q\] are mathematical statements, then in order to show that the statement “ \[p\] and \[q\]” is true, the following steps are followed.

• Step-\[1\]- Show that the statement \[p\] is true.

• Step-\[2\] - Show that the statement \[q\] is true.

• Rule \[2\]:

• Statements with “Or”.

• If p and q are mathematical statements, then in order to show that the statement “ \[p\] or \[q\]” is true, one must consider the following.

• Case \[1\]- By assuming that \[p\] is false, show that \[q\] must be true.

• Case \[2\]- By assuming that \[q\] is false, show that \[p\] must be true.

• Rule \[3\]:

• Statements with “If-then”.

• In order to prove the statement “if \[p\] then \[q\] ” we need to show that any one of the following cases is true.

• Case \[1\]- By assuming that \[p\] is true, prove that \[q\] must be true.(Direct method)

• Case \[2\]- By assuming that \[q\] is false, prove that \[p\] must be false.(Contrapositive method)

• Rule \[4\]:

• Statements with “if and only if ”

• In order to prove the statement “\[p\] if and only if \[q\] ”, we need to show.

• If \[p\] is true, then \[q\] is true and If \[q\] is true, then p is true

• By Contradiction

• To check whether a statement \[p\] is true, we assume that \[p\] is not true that is $\neg q$  is true.

• Then, we arrive at a final result that contradicts our assumption. Therefore, we conclude that \[p\] is true.

• The method involves giving an example of a situation where the statement is not valid.

• Such an example is called a counter example.

Class 11 Maths Notes of Mathematical Reasoning

The fact is that it requires one to thoroughly understand the concept and know the exact formulae to solve a given problem effortlessly. This could typically be achieved using the quick revision notes that make grasping maths all easy. So, let’s get to know what you will find in the maths class 11 mathematical reasoning notes.

The Maths Class 11 Mathematical Reasoning Notes are also available in a PDF format on the official website of Vedantu. You can download the entire Ch 14 Class 11 Maths Revision Notes on your device and then access them offline anytime. These PDFs can also be printed for further ease of use. Having Mathematical Reasoning Class 11 Notes so handy is very convenient for a quick revision during stressful exam times.

Class 11 Revision Notes Mathematical Reasoning 

Logic - Logic deals with studying formulas for reasoning, which is based on statements or propositions. Mathematical reasoning is broadly categorized into two types: 

• Inductive Reasoning - This is based on generalized statements and is a non-rigorous method of reasoning. Here, we check the validity of the statement against a set of rules. An example of inductive reasoning is:

• A man’s neighbours cat hisses at him regularly.

• At the pet store, all the cats hiss at that man.

• Hence it can be concluded that all cats hate that man.

• Deductive Reasoning - In this rigorous form of reasoning, statements are considered true only if the assumptions they are based on are true. In this, we start with a premise (a few general ideas) and then apply them to a specific situation. Some examples of deductive reasoning are:

• All humans possess muscles.

• All muscles are made of living tissues.

• With the above two statements, it can be deduced that all humans are made of living tissues.

Introduction to Important Topics in Class 11 Revision Notes Mathematical Reasoning

1. Statements

A sentence which is either true or false, but not both concurrently is known as a statement.

Points to Remember

No sentence can be termed as statement if :-

• It has an exclamation.

• It is in the form of a question.

• It is a request or an order

2. Simple Statements

A statement is said to be simple if it cannot be split down into 2 or more statements.

3. Compound Statements

A statement is said to be a compound statement if it is made up of two or more simple statements.

4. Connectives

The terms that connect or change simple statements to form compound statements or new statements are known as connectives.

5. Conjunction

If two simple statements A and B are connected by the term ‘and’, then the resulting compound statement “a and b” is known as a conjunction of a and b. It is mathematically written in symbolic form as “a ∧ b”.

Points to Remember

The statement a ∧ b has the truth value T (true) in case of both a and b have the truth value T.

The statement a ∧ b has the truth value F (false) in case of either a or b or both have the truth value F.

6. Disjunction

If two simple statements a and b are joined by the term ‘or’, then the resulting compound statement “a or b” is known as the disjunction of a and b, It is mathematically written in symbolic form as “a ∨ b”.

Points to Remember

The statement a ∨ b holds the the truth value F in case of both a and b have the truth value F.

The statement a ∨ b holds the truth value T in case either a or b or both have the truth value T.

7. Negation

A postulation that a statement fails or refutes of a statement is known as the negation of the statement. The negation of a statement ‘A’ in symbolic form is written as “~a”.

Points to Remember

~a holds the truth value T only in case p has truth value F.

~a holds the truth value F only in case a has truth value T.

8. Negation of Conjunction

The negation of a conjunction a ∧ b is said to be the disjunction of the negation of ‘a’ and the negation of ‘b’.

Correspondingly we write 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 negation of ‘a’ and the negation of ‘b’.

Correspondingly, we write it as ~(a ∨ b) = ~a ∧ ~b.

10. Negation of Negation

The statement itself is what we call Negation of negation of a statement.

Correspondingly, we write it as ~(~a) = a

11. The Conditional Statement

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

12. Contrapositive of Conditional Statement

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

13. Converse of a Conditional Statement

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