Maths Revision Notes for Mathematical Reasoning

Mathematical reasoning is an important part of most competitive exams and olympiads today. In Mathematical reasoning, we are given a hypothesis and we analyse it. Some common concepts related to Mathematical Reasoning are a statement, conjunction, disjunction, inductive reasoning, deductive reasoning, simple statements, compound statements, fallacy, etc. Many times students get confused about these topics as they are quite different from what they have studied in Maths all the time. Mathematics seems like a subject that is based on formulas and theorems, but mathematical reasoning helps students become familiar with logical and critical thinking.

To help you master the concepts of Mathematical reasoning and prepare for competitive exams like JEE, NEET, and olympiads, the askIITians Maths experts have created these free revision notes. They include pointwise explanations for various mathematical reasoning concepts so that you can refer to them while solving mathematical reasoning problems. You must study these notes carefully as they will help you learn how to solve higher-order thinking questions easily.

Some benefits of Online Revision Notes for Mathematical Reasoning:

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Free Revision Notes for Mathematical Reasoning

A sentence is called a mathematically acceptable statement if it is either true or false but not both.
A sentence is neither imperative nor interrogative nor exclamatory.
A declarative sentence containing variables is an open statement if it becomes a statement when the variables are replaced by some definite values.
A compound statement is a statement that is made up of two or more statements. Each of these statements is termed to be a compound statement.
The compound statements are combined by the word “and” (^) the resulting statement is called conjunction denoted as p ∧ q.
The compound statement with “And” is true if all its component statements are true.
The following truth table shows the truth values of p ∧ q ( p and q) and q ∧ p ( q and p):

Truth Table (p ∨ q, q ∨ p)
p	q	p ∧ q	q ∧ p
T	T	T	T
T	F	F	F
F	T	F	F
F	F	F	F
Rule: p ∧ q is true only when p and q are true

Compound statements p and q are combined by the connective ‘OR’ (∨) then the compound statement denoted as p ∨ q so formed is called a disjunction.??

Truth Table (p v q, q v p)
p	q	p ∨ q	q ∨ p
T	T	T	T
T	F	T	T
F	T	T	T
F	F	F	F
Rule: p ∨ q is false only when both p and q are false.

The denial of a statement is called the negation of the statement. The truth table for the same is given below:

Truth Table (~p)
p	~p	~ (~p)
T	F	T
F	T	F
Rule: ~ is true only when p is false

Negation is not a binary operation, it is a unary operation i.e. a modifier.
There are three types of implications:
“If ….... then”
“Only if”
“If and only if”
“If …. then” type of compound statement is called a conditional statement. The statement ‘if p then q’ is denoted by p → q or by p ⇒ q. p → q also means:
p is sufficient for q
q is necessary for p
p only if q
p leads to q
q if p
q when p
if p then q
Truth table for p → q

Truth Table (p → q, q → p)
p	q	p → q	q → p
T	T	T	T
T	F	F	T
F	T	T	F
F	F	T	T
Rule: p → q is false only when p is true and q is false.

If and only if the type of compound statement is called biconditional or equivalence or double conditional. represented as p ⇔ q or p ↔ q, it means
p is a necessary and sufficient condition for q
q is a necessary and sufficient condition for p
If p then q and if q then p
q if and only if p
Truth table for p ↔ q or q ↔ p

Truth Table (p ↔ q, q ↔ p)
p	q	p ↔ q	q ↔ p
T	T	T	T
T	F	F	F
F	T	F	F
F	F	T	T
Rule: p ↔ q is true only when both p and q have the same truth value.

The contrapositive of p → q is ~ q → ~ p.
Converse of p → q is q → p.
Truth table for p → q

Truth Table (p → q)
p	q	p → q	~q → ~p (Contrapositive)	q ↔ p (Converse)
T	T	T	T	T
T	F	F	F	T
F	T	T	T	F
F	F	T	T	T

The compound statements which are true for any truth-value of their components are called tautologies.
The truth table for a tautology ‘p ∨ ~p”?, p being a logical statement

Truth Table (p ∨ ~p)
p	~p	p ∨ ~p
T	F	T
F	T	T

The negation of tautology is a fallacy or a contradiction. The truth table for ‘p ∧ ~ p” which is a fallacy, p being a logical statement is given below

Truth Table (p ∧ ~p)
p	~p	p ∧ ~p
T	F	F
F	T	F

Important points on tautology and fallacy:
p ∨ q is true if at least one of p and q is true
p ∧ q is true if both p and q are true
A tautology is always true
A fallacy is always false.
?Statements satisfy the following laws:
Idempotent Laws: If p is any statement then p ∨ p = p and p ∧ p = p
Associative Laws: If p, q, r are any three statements, then p ∨ (q ∨ r) = (p ∨ q) ∨ r and p ∧ (q ∧ r) = (p ∧ q) ∧ r
Commutative Laws: If p, q are any two statements, then p ∨ q = q ∨ p and p ∧ q = q ∧ p
Distributive Laws: If p, q, r are any three statements then p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) and p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r)
Identity Laws: If p is any statement, t is tautology and c is a contradiction, then p ∨ t = t, p ∧ t = p, p ∨ c = p and p ∧ c = c
Complement Laws: If t is tautology, c is a contradiction and p is any statement then p ∨ (~p) = t, p ∧ (~p) = c, ~t = c and ~c = t
Involution Law: If p is any statement, then ~(~p) = p
De-Morgan’s Law: If p and q are two statements then ~(p ∨ q) = (~p) ∧ (~q) and ~(p ∧ q) = (~p) ∨ (~q)

Maths Revision Notes for Mathematical Reasoning FAQs

Is mathematical reasoning important for JEE?

Yes, mathematical reasoning questions are often asked in JEE Main. However, it is not included in the syllabus for JEE Advanced.

What is mathematical reasoning?

Mathematical reasoning is a branch of mathematics where you use Mathematical skills and logic to prove that the given statements are true. Mathematical reasoning is of two types, inductive reasoning and deductive reasoning.

How can askIITians help me in preparing Mathematical Reasoning concepts for entrance exams?

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Where can I find Maths revision notes for JEE?

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