Statements in Mathematical Reasoning

Mathematical Reasoning

Mathematical reasoning is the concept in Mathematics which deals with finding the truth values of any Mathematical statements.  The principle of Mathematical reasoning is generally used to analyze the conceptual logical thinking capacity of an individual in competitive examinations and eligibility tests. Mathematical reasoning questions are extremely interesting and successfully stirs up the rational thinking of the human brain. There are various kinds of statements in Mathematical reasoning and the operations that are performed on those statements. 

What is a Mathematical Reasoning Statement?

A Mathematical statement is a statement written in such a way that it can either be true or false but can never be both true and false simultaneously. 

For the clear understanding of “What is a Mathematical Reasoning Statement?’, let us consider two statements. 

Statement 1: “Sum of any two prime numbers is even”.

Statement 2: “Every square is a rectangle”.


In statement 1, it is given that when two prime numbers are added, the resultant number is an even number. Prime numbers are those numbers which are divisible by 1 and itself only. If we consider two prime numbers 2 and 3 as examples, the sum of 2 and 3 = 5 and sum of 2 and 2 = 4. ‘5’ is an odd number whereas ‘4’ is an even number. So, we can conclude that the sum of any two prime numbers can be either even or odd. So the statement1 may be true or false at the same time. So it is not a statement according to Mathematics.

In statement 2, it is mentioned that every square is a quadrilateral. A quadrilateral is a polygon with 4 sides, 4 angles and 4 vertices. A square also has 4 sides, 4 angles and 4 vertices. So, it is always true forever that all squares are quadrilaterals. Because the statement is completely true at an instance, it is a Mathematical statement. 

Types of Math Reasoning Statements:

Math reasoning statements are of 3 types namely Simple statements, compound statements and conditional statements.

Simple Statements:

Simple statements are those direct logical statements which do not contain any modifiers and cannot be further broken into two or more simpler statements. 

Example: “Earth is a planet”. This statement can be considered as a Math reasoning statement because it is true always. It is a simple statement because it cannot be broken down into further simple statements. 

Compound Statements:

Compound statements are the Mathematical statements that can be broken down into two or more simple statements. In other words, two or more simple statements are joined together with connectives to form a compound statement. 

Example: “Earth is a planet and has a natural satellite”. 

The above statement is a compound statement because it can be broken down into two simple statements as “Earth is a planet” and “Earth has a natural satellite”. These two simple statements are joined together ‘and’ which is a connective. 

Connectives used in Compound Statements:

There are three types of connectives that are used to combine simple statements to form a compound statement. Connectives are simple English words used to connect the statements. 

The Three Types of Connectives are:

1. Conjunction: A conjunctive statement is the compound statement which is obtained by connecting two simple statements with the connective ‘and’. The conjunction of two statements ‘p’ and ‘q’ is given as “p and q” or “p ∧ q”.

Example: The conjunction of “Time is money” and “Money is time” is given as “Time is money and money is time”.


2. Disjunction: A disjunctive statement is the compound statement that is obtained by joining two or more statements with a connecting word ‘or’. The disjunction of two statements ‘p’ and ‘q’ is given as “p or q” or “p ∨ q”.

Example: The disjunction of two statements “A natural number is an odd number” and “A natural number is an even number” is given as “A natural number is an odd number or an even number”. 


3. Negation: Negation of a statement is the statement representing denial of any statement. The word ‘not’ is used to negate a statement. Though negation does not join two statements, it gives the negative of a statement. Negation of any statement ‘p’ is given as “not p” and is symbolically represented as “~p”. 

Example: Consider a statement “7 is a prime number”. The negation of this statement is given as “7 is not a prime number”. 

Mathematical Reasoning Formulas used in Compound Statements:

If ‘p’ and ‘q’ are two Mathematical statements, then important Mathematical reasoning formulas are as follows.

  • p ∧ q represents the conjunction of two statements indicating ‘p and q’.

  • p ∨ q represents the disjunction of two statements indicating ‘p or q’.

  • ~p represents the negation of a statement and indicates ‘not p’.

  • The negation of a conjunction statement is given as the conjunction of negation of two statements.

~ (p ∧ q) = ~ p ∧ ~ q  

  • The negation of a disjunction statement is given as the disjunction of negation of both the statements. 

~ (p ∨ q) = ~ p ∨ ~ q

Conditional statements:

Conditional statements are the compound statements in which the truth value of one statement depends on the truth value of the other statement. i.e. the second statement is true only if the first statement is true. A conditional statement with ‘p’ and ‘q’ as simple statements can be symbolically represented as “p → q” which represent “p if so q”. In this statement, p is called as ‘antecedent’ and q is called the conclusion. 

Example: If there is no government holiday, then the man is available. 

In the above example, “There is no government holiday” is the condition to be checked to find whether “The man is available”. So, the former is the antecedent and the latter is the conclusion. 


Inverse, Converse and Contrapositive of Conditional statements:

Inverse: 

Inverse of a conditional statement indicates the statement representing the negation of antecedent and negation of conclusion expressed as a conditional statement.

Let an example of a conditional statement be  “If a number is less than zero, then it is negative”. 

The inverse of this statement is given as “If a number is not less than zero, then it is not negative”. 

Converse: 

Converse of a conditional statement is represented by swapping the antecedent and conclusion. 

For example the converse of the statement “If it is night, then we have dinner” is written as “If we have dinner, then it is night”. 

In the above example, it is clearly indicated that the term ‘dinner’ is used only for what we eat during the night. 

Contrapositive:

Contrapositive of a conditional statement is another conditional statement which represents the negation of conclusion of the first statement as the antecedent of the second and the negation of the antecedent of the first statement as the conclusion of the second.

Let us consider a conditional statement “If he knows swimming, then he can cross the river”. The contrapositive of this statement is written as “If he cannot cross the river, then he does not know swimming”. 

Mathematical Reasoning Formulas of Conditional Statements:

  • In any of the mathematical reasoning questions, a conditional statement is represented in “if then” form as p → q where ‘p’ is the antecedent and ‘q’ is the conclusion.

  • Inverse of a conditional statement p → q is given as ~ p → ~ q.

  • Converse of a conditional statement p → q is given as q → p.

  • Contrapositive of a conditional statement p → q is given as ~ q → ~ p.

Mathematical Reasoning Questions:

Consider two statements represented as ‘a’ and ‘b’ where a = 2 + 3 = 6 and b = I am mad. Write a conditional statement indicating a → b. Write the inverse, converse and contrapositive of a → b.


Solution:

The conditional statement a → b is written as

“If 2 + 3 = 6, then I am mad”.

Inverse of the above statement is: 

“If 2 + 3 ≠ 6, then I am not mad”.


Converse is written as:

“If I am mad, then 2 + 3 = 6”.


Contrapositive of the conditional statement is:

“If I am not mad, then 2 + 3 ≠ 6”.


Fun Facts:

  • When we apply negation to the negation of any statement, the result will be the statement itself. i.e. ~ ( ~p ) = p.

  • Phrases such as “for every” and “there exist” are used in Mathematical reasoning statements. They are called quantifiers. 

FAQ (Frequently Asked Questions)

1. What are Biconditional Statements?

Biconditional statements are those Mathematical statements which have two conditions which are mutually dependent. Biconditional statements are always compound statements. It can be represented as “if and only if” statements. A biconditional statement is symbolically represented as p ↔️ q where p and q are simple statements. 

Example: A triangle is a right triangle if and only if one of the three angles of the triangle is a right angle.

The above example indicates that the triangle is a right triangle only when one angle is 900. To be more precise, the triangle will no longer be called a right triangle, if more than or less than one of the three angles is equal to 900.

2. What is Mathematical Reasoning? Why is it Important?

  • Mathematical reasoning is a concept used in Mathematics to enhance the problem solving techniques of the students rationally. 

  • This concept enables the student to analyze the problem and choose an appropriate method to solve the problem.

  • A variety of real life Mathematical problems can be solved through the logic and critical thinking developed via Mathematical reasoning.

  • It systemizes the problem solving methodology.

  • It also helps to modulate a Mathematical problem into a computer program with suitable logical operators.

  • It convenes the Mathematicians to explain the mathematical concepts either verbally or in written form.