Statements In Mathematical Reasoning 

The concept of mathematical reasoning helps in deciding whether the given statement is correct or wrong. It is mainly dependent upon the statement. We can define mathematical reasoning as the logic used to know either the statement is true or false. This concept is widely used in competitive examinations as it helps to test the knowledge and thinking capacity of the brain of each individual.

WHAT DO YOU MEAN BY MATHEMATICAL REASONING STATEMENTS?

Statements are the fundamental factors in mathematical reasoning. The most important condition for a statement is that it can only be a true statement or a false statement. But it must not be true and false in the same instant.

Let’s consider an example to understand this well.

STATEMENT: Asia is a continent.

In the above statement, Asia is truly a continent. So the statement is true. Hence, it is considered a mathematical statement.

STATEMENT: The product of two irrational numbers is rational

The product of two irrationals may be rational sometimes and also irrational in some cases. The given statement is both true and false at once. Hence, it can’t be considered a mathematical statement.

TYPES OF MATHEMATICAL REASONING

1. Inductive type reasoning
2. Deductive type reasoning

Inductive reasoning is obtained by one’s observations. These observations give you a conclusion on your consideration and help in creating generalizations. However, your conclusion may or may not be true all the time.

Whereas, deductive reasoning is a contradiction to inductive reasoning. In deductive reasoning, we use general principles to conclude. It is mostly applied the reasoning in many fields. Example: marketing, etc.

TYPES OF MATHEMATICAL REASONING STATEMENTS

• SIMPLE STATEMENTS
  
  Simple statements are direct statements. These statements are free of logic. These are single statements that cannot be separated into two simpler statements.
  
  EXAMPLE: Mango is a fruit.
  
  The above statement is simple and direct. It is a true mathematical statement and it cannot be separated.


• COMPOUND STATEMENTS
  
  The combination of two or more simple statements is called compound statements. These statements can be separated into simple statements.
  
  EXAMPLE: India is a country with the highest population rate.
  
  The statement given above can be separated into two simple statements as “India is a country” and “India has the highest population rate.
  
  Note that both the statements are connected.


• CONDITIONAL STATEMENTS
  
  Let us consider two statements A and B.
  
  If statement A is shown to be true, then statement B will be true. And if statement B is shown to be false, then statement A will be false. These types of statements are known to be conditional statements.

EXAMPLE:

Statement 1: A school is a place of learning.

Statement 2: Everyone has to go to school.

Since school is a place of learning, everyone has to go to school for learning. So, statement 1 is proved to be true. Therefore, statement 2 is also true.

Note that both of the statements are dependent on each other.

CONCLUSION OF MATHEMATICAL STATEMENTS

To find out the result for the given statements, we use some methods which help in the correct conclusion of the statement.

1. Negative statement method
   
   In this method, simply oppose the given statement. If the given statement is true then opposition of the statement is false and vice – versa.
   
   EXAMPLE: The sum of 3 and 6 is 9 Here, the negative statement is that the sum of 3 and 6 is not 9.
   
   Since, we know that 3 + 6 = 9. Therefore, our negative statement is false and the given statement is a true mathematical statement.


2. Method of contradiction
   
   In this method, contradict the given statement and prove that the contradiction is false.
   
   EXAMPLE: 4/5√3 is irrational
   
   SOLUTION: Let us assume, to the contrary that 4/5√3 is rational.

But this contradicts the fact that √3 is irrational.

Hence,4/5√3 is an irrational number.