Mathematical Reasoning Class 11

If you are in class 11 and have opted for Maths as one of your core subjects then you must be feeling the hike in difficulty in comparison with older classes. Introduction to different branches of Mathematics like calculus, number theory, vectors, numerical analysis, and algebra occurs in this particular class only. Mathematical Reasoning class 11 is one such topic which concentrates on the theoretical part of Maths. Read on to know intricate details about this chapter!

Introduction to Mathematical Reasoning Class 11

The principle of Mathematical Reasoning is a branch of maths where you verify the truth values of the provided statements. Logic is the topic, which deals with reasoning methods. It offers rules to prove the theorem by finding out the validity of a known argument. Let us know what reasoning, and the concepts in mathematics are and be familiar with how to resolve questions easily. 

The statement is the basic unit of Mathematical Reasoning. A statement is an assured sentence that each sentence is true or false but cannot be both. If a statement is true then it is called a valid statement or else, it is called an invalid statement. Statements are represented by small letters like p, q, r, and more.

Example: ‘Addition of two prime numbers is even.’
The provided statement can be either true or false because the addition of two prime numbers can be an even or odd number. The reasoning for this statement is not acceptable mathematically, as the sentence is uncertain. So this sentence is accepted only when it is changed to ‘Either true or false, but cannot be both at the same instance.’ So, the basic unit required for Mathematical Reasoning is called a statement and hence the definition.

Types of Reasoning

Reasoning can be of 2 types in mathematical terms.

• Inductive Reasoning
• Deductive Reasoning

The other types of reasoning include abductive induction, backward induction, critical thinking, counterfactual thinking, and intuition. These 7 types of reasoning are mainly used to conclude. In this article, we have discussed the two main types – i.

e., inductive and deductive reasoning.

Inductive Reasoning is a method used to validate the statement that is verified with a definite set of rules after that generalizing. The inductive reasoning concept is employed for the principle of mathematical induction and is not considered in geometrical proofs, as it is generalized. Follow the example shown below to understand inductive reasoning better.

Example
Statement: The cost of supplies is Rs 100 and the labour cost to manufacture the item is Rs 50. The selling price of the item is Rs 500.
Reasoning: It can be said that from the given statement, the item will offer a good profit by selling it.

The deductive Reasoning principle is exactly contradictory to that of the principle of induction. In deductive reasoning, based on the previous facts for a given statement, you will conclude. The given example shown below will assist in knowing deductive reasoning concepts better.

Example
Statement: Pythagoras’ Theorem holds good for right-angled triangles.
Reasoning: If triangle ABC is a right-angled triangle, then it follows Pythagoras’ Theorem.

Types of Reasoning Statements

The 3 main types of Reasoning Statements are

• Simple Statements
• Compound Statements
• If-then Statements

Also Read: 3D Geometry Class 11 Notes

Simple Statements -Simple statements are direct sentences that don’t need much reasoning and are simple to solve. These statements are declarative enough to decide true or false.

Example
Statement: The sun sets in the west.
In the above statement, there is no modifier, and hence it can be concluded as true.

Compound Statements – Compound statements are made up of combining two or more statements with the help of definite connectives. The connectives used to club statements are ‘and’, ‘or’ and more. With the aid of these statements, the mathematical deduction concept can be executed easily. Follow the example for better understanding.

Example
Statement 1: Square of 7 is 49
Statement 2: 7 is also an odd number
Compound Statement: A square of 7 is 49, and 7 is also an odd number.

Example
Compound Statement: A rectangle has four sides, and the sum of the interior angles of a rectangle is 360°.
Statement 1: A rectangle has four sides
Statement 2: The sum of the interior angles of a rectangle is 360°
From the above two examples ‘and’ is used as a connective to form the statements.

If-then Statements – Here we come across if-then statement combinations. For example, if ‘A’ is true, then ‘B’ can be verified to be true or if you prove that ‘B’ is false, then ‘A’ is also false.

Example
A: 6 is a multiple of 36
B: 6 is a factor of 36
Given that one of the statements that are ‘A’ is true; thus A or B is true.

Method of Deducing Mathematical Statements

For making significant deductions from the known statements, three methods are generally used.

• The negation of the Certain Statement
• Contradiction Method
• Counter Statements

The negation of the Certain Statement – In this method, you make new statements from the previous ones by denying the given statement. Particularly here you deny the provided statement and express it as a fresh statement. Go through the below-shown example to understand the concept.

Statement 1: ‘Sum of cubes of three natural numbers is positive.’
Now if you negate the above statement, then it will be.
Statement 2: ‘Sum of cubes of three natural numbers is not positive.’
From the above example, “not” is used to reject the statement provided. For the conclusion that if Statement 1 is mathematically acceptable, then the rejection of Statement 1 (Represented by Statement 2) is also a statement.

Contradiction Method – In this method, we believe that the known statement is false and then attempt to prove the statement is wrong.

Example 
Consider, for all integers q, if q² is odd, q is odd.
Now you have to take the rejection of the given statement and deduce it to be true.
So consider q²  to be even. So the conclusion that p is even, then q² is the product of q is also even. This statement will disagree with the possibility that q² is odd. Therefore, the proposition is true, and the possibility is false.

Counter Statements – The counter-statement is a method to prove in which areas the provided statement is not applicable.

Example
A: If P is a prime number, then P is always even.
Now, you have to come across a negative statement to make the above statement false. You know that 3 is the prime and also an odd number that is divisible by itself and 1. Hence, you can say that 3 is an odd prime number. Therefore, the given statement ‘A’ is not true for all the prime numbers and is not valid.

If You Can Pass This Simple Math Quiz, You’re a Genius!

Quantifiers and Quantified Statements

In these statements, two important symbols are used:

• The symbol ‘∀’ stands for ‘all values of.’ This is known as the universal quantifier.
• The symbol ‘∃’ stands for ‘there exists’. The symbol ∃ is known as an existential quantifier.

Quantified Statement

An open sentence with a quantifier becomes a statement, called a quantified statement.

Negation of a Quantified Statement

• ~{∀ x ∈ A : p(x) is true} = {∃ x ∈ A such that (s.t.) ~ p(x) is true}
• ~{∃x ∈ A : p(x) is true} = {∀ x ∈ A : ~ p(x) is true}

Laws of Algebra of Statements

Idempotent Laws

• p ∨ p ≡ p
• p ∧ p ≡ p

Associative Laws

• (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
• (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)

Commutative Laws

• p ∨ q ≡ q ∨ p
• p ∧ q ≡ q ∧ P

Distributive Laws

• p ∨ (q A r) ≡ (p ∨ q) ∧ (p ∨ r)
• p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

De Morgan’s Laws

• ~(p ∨ q) ≡ (~ p) ∧ (,_ q)
• ~(p ∧ q) ≡ (~ p) ∨ (~ q)

Identity Laws

• p ∧ F ≡ F
• p ∧ T ≡ p
• p ∨ T ≡ T
• p ∨ F ≡ p

Complement Laws

• p ∨ (~ p) ≡ T
• p ∧ (~ p) ≡ F
• ~ (~p) ≡ p
• ~ T ≡ F, ~ F ≡ T

NCERT Chemistry Class 11 Solutions & Syllabus

Mathematical Reasoning Class 11 NCERT PDF

Mathematical Reasoning Class 11 PPT

Mathematical reasoning from Aza Alias

Mathematical Reasoning Class 11 MCQs

1. The contra-positive of the statement if p then q is

(a) if ~p then q

(b) if p then ~q

(c) if q then p

(d) if ~q then ~p

2. Which of the following is not a statement

(a) The product of (-1) and 8 is 8

(b) All complex numbers are real numbers

(c) Today is a windy day

(d) All of the above

3. Which of the following statement is a conjunction

(a) Ram and Shyam are friends

(b) Both Ram and Shyam are friends

(c) Both Ram and Shyam are enemies

(d) None of these

4. Which of the following is a statement

(a) x is a real number

(b) Switch of the fan

(c) 6 is a natural number

(d) Let me go

5. The converse of the statement p ⇒ q is

(a) p ⇒ q

(b) q ⇒ p

(c) ~p ⇒ q

(d) ~q ⇒ p

6. The negation of the statement The product of 3 and 4 is 9 is

(a) It is false that the product of 3 and 4 is 9

(b) Product 3 and 4 is 12

(c) The product of 3 and 4 is not 12

(d) It is false that the product of 3 and 4 is not 9

7. Sentences involving variable time such as today, tomorrow, or yesterday is

(a) Statements

(b) Not statements

(c) may or may not be statements

(d) None of these

8. The converse of the statement if a number is divisible by 10, then it is divisible by 5 is

(a) if a number is not divisible by 5, then it is not divisible by 10

(b) if a number is divisible by 5, then it is not divisible by 10

(c) if a number is not divisible by 5, then it is divisible by 10

(d) if a number is divisible by 5, then it is divisible by 10

9. Which of the following is not a negation of the statement A natural number is greater than zero

(a) A natural number is not greater than zero

(b) It is false that a natural number is greater than zero

(c) It is false that a natural number is not greater than zero

(d) None of these

10. Which of the following is the conditional p → q

(a) q is sufficient for p

(b) p is necessary for q

(c) p only if q

(d) if q then p

Answers 

1. d
2. d
3. d
4. c
5. b
6. a 
7. b
8. d 
9. c 
10. c 

Mathematical Reasoning Class 11 Important Questions

1. Write the contrapositive of the given if-then statements:

(a) If a triangle is equilateral, then it is isosceles

(b) If a number is divisible by 9, then it is divisible by 3.

2. Show that the statement, p: if a is a real number such that a3 + 4a =0, then a is 0″, is true by the direct method.
3. Which of the following sentences are statements? Justify your answer.

(i) Answer this question

(ii) All the real numbers are complex numbers

(iii) Mathematics is difficult

4. Check whether the following statement is true or not: “If a and b are odd integers, then ab is an odd integer”
5. Check the validity of the following statement: “square of the integer is positive or negative.”
6. Find the component statements for the following given statements and check whether it is true or false:

(a) A square is a quadrilateral and its four sides are equal

(b) All prime numbers are either even or odd

7. Write the negation of the following statements:

(i) p: For every positive real number x, the number x-1 is also positive.
(ii) q: All cats scratch
(iii) r: For every real number x, either x>1 or x<1.
(iv) s: There exists a number x such that 0<x<1.

8. Write the negation of the given statement: All students live in dormitories.
9. Show that the following statement is true by the method of contrapositive: ‘If x is an integer and x² is even, then x is also even’.
10. Write a component statement for the following compound statements: 50 is a multiple of both 2 and 5.

CBSE Deleted Syllabus of Class 11 [All Subjects] 2021

Key Takeaways

• In Chapter 14 Mathematical Reasoning, a mathematically acceptable proposition should be either true or untrue.
• A compound statement is composed of two or simpler assertions known as component statements.
• Compound sentences are distinguished by the terms “And,” “Or,” “There exists,” and “For every.”
• “If,” “only if,” and “if and only if” are all implications.
• If p is any assertion, then p gives the negation of ∼p.
• The contrapositive of p ⇒ q is ∼ q ⇒ ∼p, its converse is q ⇒ p, and p ⇒ q along with its converse gives p if and only if q.
• You can check the validity of statements using the direct method, contrapositive method, method of contradiction, or using counterexample.

FAQs

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