Mathematical Reasoning

Mathematical Statement

Human beings were not the same, they are today. What we are today is the result of evolution we have undergone through the centuries. Distinguished from other animals due to our intellects, we humans have one more specialty that makes us evolve. Our capability to reason every statement, come what may, helps us evolve every minute. In the following chapter, we shall discuss reasoning of any Mathematical Statement!

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Introduction

Mathematical Statement

A mathematical statement forms the basis of any kind of reasoning. Before delving into the details let’s first discuss what a mathematical statement is?

Mathematical Statement

For understanding any mathematical statement we first need to recollect what maths is basically. When we solve any problem in maths our solution is either right or wrong. There is no midway to the problems! Similar is the situation with any mathematical statement. A mathematical statement is either true or false.

Any statement which is predicted to be both cannot be a mathematical statement. For understanding this we take three sentences:

  1. The first prime minister of India was a woman.
  2. Blue Whale is the largest animal on Earth.
  3. Girls are intelligent than boys.

The first statement is false while the second is true, but when we consider the third statement for some it is true while for others it is false. All girls are not intelligent than boys. So a statement which is either true or false is called a mathematical statement.  Every statement that is either true or false is said to be a mathematically accepted one, hence is called a mathematical statement.

Example 1

The sum of a and b is greater than 0

In the sentence above we are not sure the statement is true or false as the values of a and b are not known to us. If a is a positive number say +2 and b a negative number say -5 then the answer we get is -3, which makes the above sentence false.  Whereas if the value of a is 1 and b is 2 then the sentence seems true. The above sentence is not a mathematical statement.

But if we write the same as for any natural numbers a and b, the sum of a and b is greater than 0, then the sentence is a mathematical statement. Mathematical statements are generally denoted by small letters (p,q,r,…).

Forming New Statements

We can also make new statements from already existing statements. For making new sentences we take into consideration two techniques:

Negation of Statements

Denying any statement is termed as the negation of that statement. For example, let’s consider the following sentence:

  • Jaipur is a big city.

Since it is a statement we write it as:

  • p: Jaipur is a big city.

The negation of this statement is:

  • It is false that Jaipur is a big city, or
  • Jaipur is not a big city.

Now if p is a statement then the negation of that statement shall be denoted by using ∼p. This can be read as ‘not p’.

Compound Statement

Compound statements are formed by combining two or more statements using the connecting words like “and”, “or”, “if, then”, “either, or”, etc. Consider  the following example:

  • p: Jaipur is a big and beautiful city.

This statement is a combination of two statements:

  • p: Jaipur is a big city.
  • q: Jaipur is a beautiful city.

Statements and q are connected by “and”. From the above example, we come to a conclusion: each statement in a compound statement is called the component statement. This implies that a compound statement comprises two or more component statements each connected by connecting words.

 Some Important Rules of Compound Statements

  1. A compound statement which has And as the connecting word is said to be true if all its component statements are true.
  2. Any compound statement which has And as the connecting word is said to be false if all its component statements are false.
  3. A compound statement which has Or as the connecting word is said to be true if one of its component statement or both the component statements is true.
  4. Any compound statement which has ‘Or’ as connecting word is false if both the component statements are false.

Quantifiers

Quantifiers are the phrases like “For all” or “There exists”. Consider the following example:

  • There exists a triangle whose all sides are equal.

This means that there is at least one triangle whose all sides are equal. Both the quantifiers, ‘For all’ and ‘There exists’ are closely connected as both of them signify towards a condition in the statement that applies to every condition.

These special words used in mathematical statements have special meanings attached to each of them, which helps us in knowing the validity of different statements.

A mathematical statement if understood accurately takes us in the right direction of reasoning. To deduce the best of results and reach positive conclusions we need to understand statements properly as these give our reasoning power better directions.

Solved Example for You

Question: Consider the statement, Given that people who are in need of refuge and consolation are apt to do odd things, it is clear that people who are apt to do odd things are in need of refuge and consolation. This statement, of the form (PQ)(QP)is logically equivalent to people

  1. who are in need of refuge and consolidation are not apt to do odd things
  2. that are apt to do odd things if and only if they are in need of refuge and consolidation
  3. who are apt to do odd things are in need of refuge and consolidation
  4. who are in need of refuge and consolidation are apt to do odd things

Solution: Option C. People who are apt to do odd things are in need of refuge and consolidation. Given statement is “people who are in need of refuge and consolation are apt to do odd things”. It is in the form of p⇒q, where p is “in need of refuge and consolation” and q is “apt to do odd things”.

So  qp is equivalent to “people who are apt to do odd things are in need of refuge and consolation”. Therefore option C is correct.
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