Mathematical Reasoning

Introduction

Mathematical reasoning is a type of skill of critical nature that makes a person proficient in using other mathematical skills. This type of reasoning helps make sense of the subject of maths. A person can understand mathematics much more efficiently after mastering this skill. Such a person can develop solutions, describe solutions, understand the application of solutions, derive logical conclusions, and solve problems with strategies. The study material notes on mathematical reasoning will provide a fundamental understanding of its various aspects.

Types of Mathematical Reasoning

There can be various types of mathematical reasoning. However, in this mathematical Reasoning study material, we shall focus on the following two classes as they are the most useful.

Inductive reasoning:

In this type of reasoning, checking the statement’s validity takes place by using some set of rules. Afterwards, the generalisation of the particular statement happens. As such, this type of reasoning is non-rigorous in which the conception of the statements happens.

It involves making generalisations and searching for patterns. For example, inductive reasoning is used to analyse many different triangles types. Using this reasoning, one would list the characteristics that these triangles have in common. 

Deductive reasoning:

This type of reasoning is rigorous. Here, the statements are not generalised but are assumed to be true. The statements will be considered authentic if the deductions are accurate. This type of reasoning holds more importance in mathematics than inductive reasoning.

Here, a person makes a logical argument, draws conclusions, and applies generalisations to particular situations. For example, after developing an understanding of triangles, a person can generalise new triangular figures to check whether or not each figure is a triangle.

Types of statements in Mathematical Reasoning

A Mathematical statement is a statement written so that it can either be the truth or falsehood but can never be both simultaneously. The study material notes on mathematical reasoning throw light on the three types of statements:

Simple statement: 

Simple statements are those in mathematical reasoning in which the truth value is not dependent on another statement in an explicit manner. They are direct and are devoid of any modifier. For example, the statement ‘246 is an even number’.

Compound statement: 

When the combination of two or more simple statements takes place using words like ‘if and only if’, ‘if…then’, ‘and’, and ‘or’, the resulting statement is called a compound statement, for example, the statement ‘I am studying history and political science’.

If-then Statement:

If-then statements are conditional statements in which a conclusion follows a hypothesis. Such a statement would be false if the hypothesis is true, but the conclusion is wrong. Similarly, the whole statement would be incorrect if the premise is false. For example, ‘if 55% students are girls then 45% students are boys’.

Logical Connectives in Mathematical Reasoning

A Logical Connective is a symbol that facilitates the joining of two or more propositional logics. The resulting logic is dependent only on the connective’s meaning and the logic of the input. Below are the various logical connectives in this mathematical Reasoning study material.

Conjunction: 

When creating a compound statement using ‘and’, it’s called a conjunction. 

a ^ b

Disjunction: 

Disjunction in mathematical reasoning is a compound statement whose creation takes place using ‘or’. 

a v b

Negation: 

This is a statement whose creation takes place using words like ‘no’ ‘not’. 

~a

Conditional statement: 

Creating such a statement takes place by connecting two simple statements using ‘if….then’.

a → b

Biconditional statement: 

A biconditional statement is created by combining two simple statements using ‘if and only if.

a ↔ b

Value of a Statement

A statement can either be:

• Correct or incorrect 
• True or false

 Truth value refers to a statement’s true or false state. The statement will be determined as ‘F’ if it is false. It will be defined as ‘T’ if the statement is true.

Example:

• ‘264 is an even number’ is T because the statement is true.
• ’61 is divisible by 24′ is F due to the false statement.

Conclusion

Mathematical reasoning is a type of skill of critical nature that builds up or enhances proficiency in all other mathematical skills. You can make sense of maths in a much better manner after gaining this skill. The two most important types of mathematical reasoning are inductive and deductive reasoning. There are three types of statements- simple statement, compound, and If-then statement. The logical connectives are conjunction, disjunction, negation, conditional statement, and biconditional statement.