Statements In Mathematical Reasoning

Mathematical reasoning is a part of mathematics where we determine the truth values of the given statements. In terms of mathematics, reasoning can be of two types

(i) Inductive Reasoning

(ii) Deductive Reasoning

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.

On the other hand in Deductive reasoning, we apply the rules of a general case to a given statement and make it true for particular statements. This is actually opposite of the principle of induction.

Consider the following Statement: The sum of two prime numbers is always even.

The given statement can either be true or false. 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.

For deducing new statements or for making important deductions from the given statements two techniques are used generally:

i) Negation of the given statement

ii) Compound Statement

Let’s take a look at both the methods one by one.

(i) 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 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 negation of 1 (denoted by 2) is also a statement.

(ii) Compound 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 statements:

Compound Statements: 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 side.

Statement 2: The sum of interior angles of a triangle is \(180^{\circ}\)..

These both statements are mathematically true. These two statements are connected using “and.”

Now it would be clear to you how to use compound form of statements and negative of a statement to deduce results. To learn more on this topic, download Byjus-the learning app.

Practise This Question

Which point is interior to ABC?