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Mathematical Reasoning: Introduction

Mathematical Reasoning is a topic covered under the syllabus of JEE-Main only, excluding JEE-Advanced exam. One question worth 4 marks is asked from this topic in JEE-Mains paper. Generally, students don’t pay much attention to this topic especially those who are targeted for JEE-Advanced. So here, we try and give you a brief overview of mathematical reasoning through this note.

Logic is the subject that deals with the principles of reasoning. Sometimes, we define logic as the science of proof.

Statements of Logical Sentences

We convey our daily views in the form of sentence which is a collection of words. This group of words is a sentence if it makes some sense.

A declarative sentence, whose truth or falsity can be decided is called a statement of a logical sentence but the sentence should not be imperative, interrogative and exclamatory. A statement is usually denoted by p,q,r or any other small alphabet.

Open Statement:

A sentence which contains one or more variables such that when certain values are given to the variables, it becomes a statement is called an open statement.

Compound Statement:

If two or more simple statements are combined by the use of words such as ‘and’, ‘or’, ‘not’, ‘if’, ‘then’, and ‘if and only if’, then the resulting statement is called a compound statement.

Mathematical Reasoning: Truth Value and Truth Table

A statement can either be ‘true’ or ‘false’ which are called truth values of a statement and these are represented by the symbols T and F, respectively.

truth table is a summary of truth values of the resulting statements for all possible assignment of values to the variables appearing in a compound statement.

The number of rows depends on their number of statements.

p

q

T T
T F
F T
F F

Logical Operations

The phrases or words which connect simple statements are called logical connectives/operations or sentential connectives or simply connectives.

  • AND Operation

A compound sentence formed by two simple sentences p and q using connective ‘and’ is called the conjunction of p and q. It is represented by p and q.

p

q

p and q

T T T
T F F
F T F
F F F
  • OR Operation

A compound statement formed by two simple sentences p and q using connectives ‘or’ is called disjunction of p and q. It is represented by p or q.

p

q

p or q

T T T
T F T
F T T
F F F
  • Negation/NOT Operation

A statement which is formed by changing the truth value of a given statement by using the word like “no”, ‘not’ is called negation of given statement. If p is a statement, then negation of p is denoted by ~p

p

~p

T F
T F
F T
F T
  • Conditional Operation

Two simple statements p and q connected by the phrase ‘if and then’ is called conditional statement of p and q. It is represented by p => q

p

q

p => q

T T T
T F F
F T T
F F T
  • Biconditional Operation

The two simple statements connected by the phrase ‘if and only if’, this is called biconditional statement. It is denoted by the symbol p <=> q

p

q

p <=> q

T T T
T F F
F T F
F F T

Implications

Students often get confused when these four terms are played in their mind: Reverse, Converse, Inverse and Contrapositive.

By definition, the reverse of an implication means the same as the original implication itself. Each implication implies its contrapositive, even intuitionistically. In classical logic an implication is logically equivalent to its contrapositive and moreover, its inverse is logically equivalent to its converse.

Consider the implication formula p => q

  • Its reverse is q <= p
  • Its converse is q => p
  • Its inverse is ~p => ~q
  • Its contrapositive is ~q => ~p

Tautology and Contradiction

The compound statement which is true for every value of its components is called tautology. For an example, ( p =>  q ) or  ( q =>  p ) is a tautology.

The compound statement which is false for every value of its components is called contradiction/fallacy. For an example, ~ { ( p =>  q ) or ( q => p )} is a fallacy.

Arithmetic 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)

Duality

Two compound statements S1 and S2 are said to be duals of each other, if one can be obtained from the other by replacing and by or and or by and. The connectives  and  are also called duals of each other.

Additional Useful Points About Mathematical Reasoning

  • If a compound statement is made up of n substatements, then its truth value will contain  rows.
  • A statement which is neither a tautology nor a contradiction is a contingency.
  • 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 which is made up of two or more statements. Each of this statement is termed to be a compound statement.
  • The compound statements are combined by the word “and” (^) the resulting statement is called a 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):
  • 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 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
  • Contrapositive of p → q is ~ q → ~ p.
  • Converse of p → q is q → p.
  • p ∨ q is true iff at least one of p and q is true
  • p ∧ q is true iff both p and q are true
  • A tautology is always true
  • A fallacy is always false.

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