Logical Reasoning

Logical reasoning in mathematics is all about using facts, rules, and structured thinking to arrive at valid conclusions. It helps you solve puzzles, justify answers, and evaluate whether statements are true or false.

 

Types of Reasoning

1. Deductive Reasoning

In deductive reasoning, we move from general rules to specific conclusions.

Example:

All prime numbers are odd (except 2). 3 is a prime number. So, 3 is odd.

2. Inductive Reasoning

In inductive reasoning, we observe patterns and make generalisations.

Example:

The first three even numbers are 2, 4, 6. It seems like even numbers go up by 2. So, the next will be 8.

 

Statements and Truth Values

A statement is a sentence that is either true or false — not both.

Examples of statements:

• 7 is an odd number (True)
• 5 is divisible by 2 (False)

Not a statement: "What is your name?" (It's a question — not true or false)

 

Logical Connectives

We use logical operators to join or modify statements:

1. Negation ($\neg p$)

Means “not”. If $p$ is true, $\neg p$ is false.

2. Conjunction ($p \land q$)

Means “and”. True only if both $p$ and $q$ are true.

3. Disjunction ($p \lor q$)

Means “or”. True if at least one of $p$ or $q$ is true.

4. Implication ($p \Rightarrow q$)

Means “if p, then q”. False only when $p$ is true and $q$ is false.

5. Biconditional ($p \Leftrightarrow q$)

Means “p if and only if q”. True when both have the same truth value.

 

Truth Tables

Truth tables help us work out whether compound statements are true or false in every possible case.

Example: Conjunction

$p$	$q$	$p \land q$
T	T	T
T	F	F
F	T	F
F	F	F

Example: Disjunction

$p$	$q$	$p \lor q$
T	T	T
T	F	T
F	T	T
F	F	F

 

Worked Example

Practice Problem