Mathematical Reasoning, Introduction, Statement (or Proposition), Negation of a Statement

Glide to success with Doorsteptutor material for UGC : Get complete video lectures from top expert with unlimited validity: cover entire syllabus, expected topics, in full detail- anytime and anywhere & ask your doubts to top experts.

Download PDF of This Page (Size: 122K)

Introduction:

In this lesson, we shall learn about some basic ideas of mathematical reasoning and the process of reasoning especially in context of mathematics.

In mathematical language, there are two kinds of reasoning. (i) Inductive reasoning and (ii) Deductive reasoning. We have already discussed the inductive reasoning in mathematical induction. Now, we shall discuss some fundamentals of deductive reasoning.

Statement (Or Proposition):

The basic unit involved in mathematical reasoning is a mathematical statement:

A sentence is called a mathematically acceptable statement if it is either true or false but not both at the same time.

If a statement is true, we say that it is a valid statement. A false statement is known as an invalid statement.

Consider the following two sentences:

Three plus four is .

Two plus three is .

When we read these sentences, we immediately decide that the first sentence is wrong and second is correct. There is no confusion regarding these. In mathematics such sentences are called statements.

Now consider the following sentence:

Mathematics is fun.

Mathematics is fun is true for those who like mathematics. But, for others, it may not be true. So, the given sentence is true or false both. Hence, it is not a statement.

Consider the following sentences:

  • Moon revolves around the Earth.

  • Every square is a rectangle.

  • The Sun is a Star.

  • Every rectangle is a square.

  • New Delhi is in Pakistan

When we read these sentences, the first, second and third sentences are true but fourth and fifth are-false sentences. Hence, each of them is a statement.

Consider the following sentences:

  • Give me a glass of water

  • Switch on the light

  • Where are you going?

  • How are you?

  • How beautiful!

  • May you live long!

  • Tomorrow is Wednesday

We cannot decide the truth value of (i), (ii), (iii), (iv), (v), (vi) and (vii). Hence, they are not statements.

Example:

Check whether the following sentences are statements. Give reasons for your answer.

(i) 12 is less than 16.

(ii) Every set is a finite set.

(iii) x + 5 = 11.

(iv) There is no rain without clouds.

(v) All integers are natural numbers.

(vi) How far is Agra form here?

(vii) Are you going to Kanpur?

(viii) All roses are white.

Solution:

(i) This sentences is true, because ( is less than ). Hence, it is a statement.

(ii) This sentence is false, because there are sets which are not finite. Hence, it is a statement.

(iii) The sentence is an open sentence. Its truth value cannot be confirmed unless we are given the value of x. Hence, it is not a statement.

(iv) It is scientifically established natural phenomenon that cloud is formed before it rains. Therefore, this sentence is always true. Hence, it is a statement.

(v) This sentence is false, because all integers are not natural numbers. So, it is a statement.

(vi) This sentence is a question (or interrogative sentence). Hence, it is not statement.

(vii) We can’t have a truth value for it. So it is not a statement.

(viii) This sentence is false, because all roses are not white. Hence, it is a statement.

Negation of a Statement:

“The denial of a statement is called the negation of the statement.”

Let us consider the statement:

: New Delhi is a city.

The negation of this statement is

It is not the case that New Delhi is a city.

Or

It is false that New Delhi is a city

Or

New Delhi is not a city.

If is statement, then the negation of is also a statement and is denoted by , and read as ‘not ’.

Example:

Write the negation of the following statements:

(i) Sum of and is .

(ii) is rational.

(iii) Australia is a continent.

(iv) The number is less than .

Solution:

(i) Sum of and is .

Sum of and is not .

(ii) is rational

is not rational

Or

It is false that is rational

(iii) Australia is a continent

Australia is not a continent

(iv) The number is less than .

The number is not less than .

Or

It is false that the number is less than .

Developed by: