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.
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
Among 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. 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 labor 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 actually 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.
How to Deduce Mathematical Statements?
For deducing new statements or for making important deductions from the given statements two techniques are generally used:
- Negation of the given statement
- Compound Statement
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.
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.”
Now it would be clear to you how to use a compound form of statements and negative of a statement to deduce results. To learn more on this topic, register at BYJU’S now and download BYJU’S- The Learning App.