Mathematical Reasoning and Statements
What is Mathematical Reasoning?
Mathematical reasoning or the principle of mathematical reasoning is a part of mathematics where we determine the truth values of the given statements. These reasoning statements are common in most of the competitive exams like JEE and the questions are extremely easy and fun to solve. Let us understand what reasoning in maths is in this article and know how to solve questions easily.
Table of Contents:
- Mathematically Accepted Statements
- Types of Mathematical Reasoning
- Types of Reasoning Statements
- Deducing Mathematical Statements
- Example Questions
Mathematically Acceptable Statements
Consider the following Statement:
“The sum of two prime numbers is always even.”
The given statement can either be true or false since the sum of two prime numbers can be either be an even number or an odd number. Such statements are mathematically not acceptable for reasoning as this sentence is ambiguous. Thus, a sentence is only acceptable mathematically when it is “Either true or false, but not both at the same time.” Therefore, the basic entity required for mathematical reasoning is a statement. This is the mathematical statement definition.
Types of Reasoning in Maths
In terms of mathematics, reasoning can be of two major types which are:
- Inductive Reasoning
- Deductive Reasoning
The other types of reasoning are intuition, counterfactual thinking, critical thinking, backwards induction and abductive induction. These are the 7 types of reasoning which are used to make a decision. But, in mathematics, the inductive and deductive reasoning are mostly used which are discussed below.
Note: Inductive reasoning is non-rigorous logical reasoning and statements are generalized. On the other hand, deductive reasoning is rigorous logical reasoning, and the statements are considered true if the assumptions entering the deduction are true. So, in maths, deductive reasoning is considered to be more important than inductive.
In the Inductive method of mathematical reasoning, the validity of the statement is checked by a certain set of rules and then it is generalized. The principle of mathematical induction uses the concept of inductive reasoning.
As inductive reasoning is generalized, it is not considered in geometrical proofs. Here, is an example which will help to understand the inductive reasoning in maths better.
- Example of Inductive Reasoning:
Statement: The cost of goods is Rs 10 and the cost of labour to manufacture the item is Rs. 5. The sales price of the item is Rs. 50.
Reasoning: From the above statement, it can be said that the item will provide a good profit for the stores selling it.
The principal of deductive reasoning is the opposite of the principle of induction. On the contrary to inductive reasoning, in deductive reasoning, we apply the rules of a general case to a given statement and make it true for particular statements. The principle of mathematical induction uses the concept of deductive reasoning (contrary to its name). The below-given example will help to understand the concept of deductive reasoning in maths better.
- Example of Deductive Reasoning:
Statement: Pythagorean Theorem holds true for any right-angled triangle.
Reasoning: If triangle XYZ is a right triangle, it will follow Pythagorean Theorem.
Types of Reasoning Statements
There are three main types of reasoning statements:
- Simple Statements
- Compound Statements
- If-Then Statements
Simple statements are those which are direct and do not include any modifier. These statements are more comfortable to solve and does not require much reasoning. An example of a simple statement is:
a: The Sun rises in the east
In this statement, there is no modifier and thus it can be simply concluded as true.
With the help of certain connectives, we can club different statements. Such statements made up of two or more statements are known as compound statements. These connectives can be “and”, “or”, etc.
With the help of such statements, the concept of mathematical deduction can be implemented very easily. For a better understanding, consider the following example:
Statement 1: Even numbers are divisible by 2
Statement 2: 2 is also an even number
These two statements can be clubbed together as:
Compound Statement: Even numbers are divisible by 2 and 2 is also an even number
Let us now find the statements out of the given compound statement:
Compound Statement: A triangle has three sides and the sum of interior angles of a triangle is 180°
The Statements for this statement is:
Statement 1: A triangle has three sides.
Statement 2: The sum of the interior angles of a triangle is 180°.
These both statements related to triangles are mathematically true. These two statements are connected using “and.”
Learn More: Tautology and Contradiction
According to mathematical reasoning, if we encounter an if-then statement i.e. ‘if a then b’, then by proving that a is true, b can be proved to be true or if we prove that b is false, then a is also false.
If we encounter a statement which says ‘a if and only if b’, then we can give reason for such a statement by showing that if a is true, then b is also true and if b is true, then a is also true.
a: 8 is multiple of 64
b: 8 is a factor of 64
Since one of the given statements i.e. a is true, therefore, a or b is true.
How to Deduce Mathematical Statements?
For deducing new statements or for making important deductions from the given statements three techniques are generally used:
- Negation of the given statement
- Contradiction Method
- Counter Statements
Let’s take a look at both the methods one by one.
Negation of the Given Statement
In this method, we generate new statements from the old ones by the rejection of the given statement. In other words, we deny the given statement and express it as a new one. Consider the following example to understand it better.
Statement 1: “Sum of squares of two natural numbers is positive.”
Now if we negate this statement then we have,
Statement 2: Sum of squares of two natural numbers is not positive.
Here, by using “not”, we denied the given statement and now the following can be inferred from the negation of the statement:
There exist two numbers, whose squares do not add up to give a positive number.
This is a “false” statement as squares of two natural numbers will be positive.
From the above discussion, we conclude that if (1) is a mathematically acceptable statement then the negation of statement 1 (denoted by statement 2) is also a statement.
Negation of Statements
In this method, we assume that the given statement is false and then try to prove the assumption wrong.
a: The derivative of y = 9x2 + sin x w.r.t x is 18x + cos x.
For proving the validity of this statement, let us say that dy/dx ≠ 18x + cos x. We know that the derivative of xn is given by n • xn−1. Therefore, the derivative of 9x2 is 18x and the derivative of sin x is given by cos x.
Therefore, d/dx (9x2 + sin x) = 18x + cos x
Hence, our assumption is wrong and the statement “a” is a valid statement.
Another method for proving validity is to use a counter statement i.e. giving a statement or an example where the given statement is not valid.
a: If x is a prime number then x is always odd.
To show that the given statement is false we will try to find a counter statement for this. We know that 2 is a prime number i.e. it is divisible by only itself and 1. Also, 2 is the smallest even number. Therefore, we can say that 2 is a prime number which is even. Hence, we can say that the statement “a” is not true for all prime numbers, therefore, the given statement is not valid.
Example Questions Using the Principle of Mathematical Reasoning
Consider the following set of statements and mention which of these are mathematically accepted statements:
i) The Sun rises in the east.
ii) New Delhi is a country.
iii) Red rose is more beautiful than a yellow rose.
When we read the first statement we can straightaway say that the first statement is definitely true and the second one is definitely false. As far as the third statement is considered it may depend upon perceptions of different people. Hence, it can be true for some people and at the same time false for others. But such ambiguous statements are not acceptable for reasoning in mathematics.
Thus a sentence is only acceptable mathematically when it is either true or false but not both at the same time. So, statement 1 and 2 are mathematically accepted statements while statement 3 is not accepted mathematically.
Question 2: The sum of three natural numbers x,y and z is always negative.
This statement is acceptable. It can never be true because all natural numbers are greater than zero and therefore the sum of positive numbers can never be negative.
Question 3: The product of three real numbers x,y and z is always zero.
In this given statement we cannot figure out if the statement is true or false. Such a sentence is not mathematically acceptable for reasoning.
Question 4: Check whether the given two statements are true with respect to each other.
a: A circle with infinite radius is a line
b: A circle with zero radii is a point
Since “a” is true and “b” is also true then both statements a and b are also true.
For two given statements a or b to be true, show that either a is true or prove that b is true i.e. if any one of the statements is true then a or b is also true.
Now it would be clear to you how to use a compound form of statements and negative of a statement to deduce results.
Video Lesson- Validity of Statements
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Frequently Asked Questions
What is Mathematical Reasoning?
Mathematical reasoning is one of the topics in mathematics where the validity of mathematically accepted statements is determined using logical and Maths skills.
Why is Mathematical Reasoning Important?
Mathematical reasoning is important as it helps to develop critical thinking and understand Maths in a more meaningful way. The concepts of reasoning not only helps the students to have a deeper understanding of the subject but also helps in having a wider perspective to logical statements.
What are the Types of Reasoning?
There are two main types of reasoning in Maths:
- Inductive reasoning
- Deductive reasoning